GEODESY 


INCLUDING 


ASTRONOMICAL    OBSERVATIONS,    GRAVITY 

MEASUREMENTS,   AND    METHOD 

OF   LEAST   SQUARES 


BY 


GEORGE    L.    HOSMER 

Associate  Professor  of  Topographical  Engineering, 
Massachusetts  Institute  of  Technology 


FIRST  EDITION 


NEW  YORK 

JOHN   WILEY   &   SONS,    INC. 

LONDON:    CHAPMAN  &  HALL,  LIMITED 

1919 


COPYRIGHT,  1919, 

BY 
GEORGE  L.  HOSMER 


Stanhope  jprcss 

?.    H.  GILSON   COMPANY 
BOSTON,  U.S.A. 


PREFACE 

In  this  volume  the  author  has  attempted  to  produce  a  text- 
book on  Geodesy  adapted  to  a  course  of  moderate  length.  The 
material  has  not  been  limited  to  what  could  be  actually  covered 
in  the  class,  but  much  has  been  included  for  the  purpose  of  giving 
the  student  a  broader  outlook  and  encouraging  him  to  pursue  the 
subject  farther.  Numerous  references  are  given  to  the  standard 
works. 

Throughout  the  book  the  aim  has  been  to  make  the  underlying 
principles  clear,  and  to  emphasize  the  theory  as  well  as  the  details 
of  field  work.  The  methods  of  observing  and  computing  have 
been  brought  up  to  date  so  as  to  be  consistent  with  the  present 
practice  of  the  Coast  and  Geodetic  Survey. 

The  chapters  on  astronomy  and  least  squares  are  included  for 
the  sake  of  completeness  but  do  not  pretend  to  be  more  than  in- 
troductions to  the  standard  works.  The  student  cannot  expect 
to  master  either  of  these  subjects  in  a  short  course  on  geodesy,  but 
must  make  a  special  study  of  each. 

The  author  desires  to  acknowledge  his  indebtedness  to  those 
who  have  assisted  in  the  preparation  of  this  book,  and  especially 
to  Professor  J.  W.  Howard  of  the  Massachusetts  Institute  of 
Technology  for  suggestions  and  criticism  of  the  manuscript;  to 
the  Superintendent  of  the  Coast  and  Geodetic  Survey  for  valuable 
data  and  for  the  use  of  many  photographs  for  illustrations;  and  to 
Messrs.  C.  L.  Berger  &  Sons  for  the  use  of  photographs  of  the 
pendulum  apparatus  and  several  electrotype  plates.  Tables  XII 
to  XVII  are  from  electrotype  plates  from  Breed  and  Hosmer's 
Principles  and  Practice  of  Surveying,  Vol.  II. 

G.  L.  H. 

CAMBRIDGE,  April,  1919. 

iii 


405416 


TABLE   OF   CONtENTS 


CHAPTER  I 
GEODESY  AND   GEODETIC   SURVEYING  —  TRIANGULATION 

ART.  PAGE 

1  Geodesy I 

2  Geodetic  Surveying I 

3  Triangulation 2 

4  Classes  of  Triangulation 2 

5  Length  of  Line 3 

6  Check  Bases 4 

7  Geometric  Figure 6 

8  Strength  of  Figure 6 

9  Number  of  Conditions  in  a  Figure 9 

10  Allowable  Limits  of  Ri  and  R* 10 

11  Reconnoissance n 

12  Calculation  of  Height  of  Observing  Tower 11 

13  Method  of  Marking  Stations 19 

14  Signals  —  Tripods .' 18 

15  Heliotropes 19 

16  Acetylene  Lights 23 

17  Towers 25 

18  Reconnoissance  for  Base  Line 28 

CHAPTER  H 
BASE  LINES 

19  Bar  Apparatus  for  Measuring  Bases 31 

20  Steel  Tapes 31 

21  Invar  Tapes 32 

22  Accuracy  Required 33 

23  Description  of  Apparatus 34 

24  Marking  the  Terminal  Points 36 

25  Preparation  for  the  Measurement 36 

26  Measuring  the  Base 36 

27  Corrections  to  Base-Line  Measurements  —  Correction  for  Grade 37 

28  Corrections  for  Alignment 38 

29  Broken  Base 38 

30  Correction  for  Temperature 39 

v 


VI  CONTENTS 

ART.  PAGE 

31  Correction  for  Absolute  Length 39 

32  Reduction  of  Base  to  Sea-Level 40 

33  Correction  for  Sag 41 

34  Tension 42 

CHAPTER  III 
FIELD  WORK  OF  TRIANGULATION  —  MEASUREMENT  OF  ANGLES 

35  Instruments  Used  in  Measuring  Horizontal  Angles 44 

36  The  Repeating  Instrument 44 

37  The  Direction  Instrument 46 

38  The  Micrometer  Microscope 48 

39  Run  of  the  Micrometer 49 

40  Vertical  Collimator 52 

41  Adjustments  of  the  Theodolite 53 

42  Effect  of  Errors  of  Adjustment  on  Horizontal  Angles 54 

43  Method  of  Measuring  the  Angles 56 

44  Program  for  Measuring  Angles 58 

45  Time  for  Measuring  Horizontal  Angles 63 

46  Forms  of  Record 64 

47  Accuracy  Required 65 

48  Reduction  to  Centre , 65 

49  Phase  of  Signal 67 

50  Measures  of  Vertical  Angles 68 

CHAPTER  IV 
ASTRONOMICAL   OBSERVATIONS 

51  Astronomical  Observations  —  Definitions 71 

52  The  Determination  of  Time 73 

53  The  Portable  Astronomical  Transit 74 

54  The  Reticle 76 

55  Transit  Micrometer 76 

56  Illumination 78 

57  Chronograph 78 

58  Circuits 81 

59  Adjustment  of  the  Transit 81 

60  Selecting  the  Stars  for  Time  Observations 83 

61  Making  the  Observations 84 

62  The  Corrections 87 

63  Level  Correction 87 

64  Pivot  Inequality 87 

65  Collimation  Correction 89 

66  Azimuth  Correction 89 

67  Rate  Correction 90 


CONTENTS  vii 

ART.  PAGE 

68  Diurnal  Aberration 90 

69  Formula  for  the  Chronometer  Correction 91 

70  Method  of  Deriving  Constants  a  and  c,  and  the  Chronometer  Correction, 

Ar 92 

71  Accuracy  of  Results 97 

72  Determination  of  Differences  in  Longitude 97 

73  Observations  by  Key  Method x 99 

74  Correction  for  Variation  of  the  Pole 101 

75  Determination  of  Latitude 101 

76  Adjustments  of  the  Zenith  Telescope 103 

77  Selecting  Stars 103 

78  Making  the  Observations 104 

79  Formula  for  Latitude 105 

80  Calculation  of  the  Declinations 106 

81  Correction  for  Variation  of  the  Pole ! 106 

82  Reduction  of  the  Latitude  to  Sea-Level 107 

83  Accuracy  of  the  Observed  Latitude 109 

84  Determination  of  Azimuth no 

'  85   Formula  for  Azimuth in 

86  Curvature  Correction 112 

87  Correction  for  Diurnal  Aberration 112 

88  Level  Correction 113 

89  The  Direction  Method •. 113 

90  Method  of  Repetition 116 

91  Micrometric  Method 118 

92  Reduction  to  Sea-Level  —  Reduction  to  Mean  Position  of  the  Pole ....  120 

CHAPTER  V 
PROPERTIES  OF  THE  SPHEROID 

93  Mathematical  Figure  of  the  Earth 122 

94  Properties  of  the  Ellipse 123 

95  Radius  of  Curvature  of  the  Meridian 125 

96  Radius  of  Curvature  in  the  Prime  Vertical 126 

97  Radius  of  Curvature  of  Normal  Section  in  any  Azimuth 128 

98  The  Mean  Value  of  Ra 131 

99  Geometric  Proofs 132 

100  Length  of  an  Arc  of  the  Meridian 134 

101  Miscellaneous  Formulas 135 

102  Effect  of  Height  of  Station  on  Azimuth  of  Line 136 

103  Refraction . 139 

104  Curves  on  the  Spheroid  —  The  Plane  Curves 139 

105  The  Geodetic  Line 140 

106  The  Alignment  Curve 143 

107  Distance  between  Plane  Curves 144 


V1U  CONTENTS 


CHAPTER  VI 
CALCULATION   OF  TRIANGULATION 

ART.  PAGE 

108  Preparation  of  the  Data 147 

109  Solution  of  a  Spherical  Triangle  by  Means  of  an  Auxiliary  Plane  Triangle  149 

no   Spherical  Excess 149 

in   Proof  of  Legendre's  Theorem 150 

112  Error  in  Legendre's  Theorem 152 

113  Calculation  of  Spheroidal  Triangles  as  Spherical  Triangles 152 

114  Calculation  of  the  Plane  Triangle 153 

115  Second  Method  of  Solution  by  Means  of  an  Auxiliary  Plane  Triangle. ..  154 


CHAPTER  VII 
CALCULATION   OF   GEODETIC   POSITIONS 

116  Calculation  of  Geodetic  Positions 158 

117  The  North  American  Datum 159 

118  Method  of  Computing  Latitude  and  Longitude 160 

119  Difference  in  Latitude 160 

120  Difference  in  Longitude 164 

121  Forward  and  Back  Azimuths 166 

122  Formulae  for  Computation 168 

123  The  Inverse  Problem 170 

124  Location  of  Boundaries 171 

125  Location  of  Meridian 172 

126  Location  of  Parallel  of  Latitude 172 

127  Location  of  Arcs  of  Great  Circles ' 174 

128  Plane  Coordinate  Systems 174 

129  Calculation  of  Plane  Coordinates  from  Latitude  and  Longitude 175 

130  Errors  of  a  Plane  System 180 

131  Adjusting  Traverses  to  Triangulation 183 


CHAPTER  VIII 
FIGURE  OF  THE  EARTH 

132  Figure  of  the  Earth 185 

133  Dimensions  of  the  Spheroid  from  Two  Arcs 187 

134  Oblique  Arcs IQO 

135  Figure  of  the  Earth  from  Several  Arcs 191 

136  Principal  Determinations  of  the  Spheroid 193 

137  Geodetic  Datum 195 

138  Determination  of  the  Geoid 196 


CONTENTS  IX 

ART.  PAGE 

139  Effect  of  Masses  of  Topography  on  the  Direction  of  the  Plumb  Line. . .  197 
—  Laplace  Points 201 

140  Isostasy  —  Isostetic  Compensation 202 


CHAPTER  IX 
GRAVITY  MEASUREMENTS 

141  Determination  of  Earth's  Figure  by  Gravity  Observations 206 

142  Law  of  the  Pendulum 206 

143  Relative  and  Absolute  Determinations 206 

144  Variation  of  Gravity  with  the  Latitude 207 

145  Clairaut's  Theorem 210 

146  Pendulum  Apparatus 211 

147  Apparatus  for  Determining  Flexure  of  Support  216 

148  Methods  of  Observing 219 

149  Calculation  of  Period 221 

150  Corrections 222 

151  Form  of  Record  of  Pendulum  Observations .  .• 232 

152  Calculation  of  g 233 

153  Reduction  to  Sea-level : . . . .  233 

154  Calculation  of  the  Compression 235 


CHAPTER  X 
PRECISE  LEVELING  —  TRIGONOMETRIC  LEVELING 

x55   Precise  Leveling 237 

156  Instrument 240 

157  Rods 240 

158  Turning  Points 241 

159  Adjustments 241 

160  Method  of  Observing 242 

161  Computing  the  Results 243 

162  Bench  Marks 245 

163  Sources  of  Error 245 

164  Datum 249 

165  Potential 250 

166  The  Potential  Function 250 

167  The  Potential  Function  as  a  Measure  of  the  Work  Done 251 

168  Equipotential  Surfaces 252 

169  The  Orthometric  Correction 254 

170  The  Curved  Vertical .- . .  256 

171  Trigonometric  Leveling 257 


X  CONTENTS 

ART.  PAGE 

172  Reduction  to  Station  Mark 257 

173  Reciprocal  Observations  of  Zenith  Distances 258 

174  When  Only  One  Zenith  Distance  is  Observed •. 261 

CHAPTER  XI 
MAP  PROJECTIONS 

175  Map  Projections 265 

176  Simple  Conic  Projection 265 

177  Bonne's  Projection / 267 

178  The  Polyconic  Projection 268 

179  Lambert's  Projection 271 

180  The  Gnomonic  Projection 273 

181  Cylindrical  Projection 274 

182  Mercator's  Projection 274 

183  Rectangular  Spherical  Coordinates 278 

CHAPTER  XII 

APPLICATION  OF  METHOD  OF  LEAST  SQUARES  TO  THE  ADJUST- 
MENT OF  TRIANGULATION 

184  Errors  of  Observation 279 

185  Probability 280 

186  Compound  Events 280 

187  Errors  of  Measurement  —  Classes  of  Errors 281 

188  Constant  Errors 281 

189  Systematic  Errors 281 

190  Accidental  Errors 281 

191  Comparison  of  Errors 281 

192  Mistakes 283 

193  Adjustment  of  Observations < 283 

194  Arithmetical  Mean 284 

195  Errors  and  Residuals 284 

196  Weights 284 

197  Distribution  of  Accidental  Errors 285 

198  Computation  of  Most  Probable  Value 290 

199  Weighted  Observations 291 

200  Relation  of  h  and  p 291 

201  Formation  of  the  Normal  Equations 293 

202  Solution  by  Means  of  Corrections 293 

203  Conditioned  Observations 294 

204  Adjustment  of  Triangulation 295 

205  Conditions  in  a  Triangulation 296 

206  Adjustment  of  a  Quadrilateral 297 


CONTENTS  xi 

ART.  PAGE 

207  Solution  by  Direct  Elimination 302 

208  Gauss's  Method  of  Substitution 302 

209  Checks  on  the  Solution 304 

210  Method  of  Correlatives 304 

211  Method  of  Directions 309 

212  Adjusting  New  Triangulation  to  Points  Already  Adjusted 310 

213  The  Precision  Measures 314 

214  The  Average  Error 316 

215  The  Mean  Square  Error 316 

216  The  Probable  Error 317 

217  Computation  of  the  Precision  Measures,  Direct  Observations  of  Equal 

Weight : 319 

218  Observations  of  Unequal  Weight 321 

219  Precision  of  Functions  of  the  Observed  Quantities 322 

220  Indirect  Observations 324 

221  Caution  in  the  Application  of  Least  Squares 325 


GEODESY 


CHAPTER  I 

GEODESY  AND    GEODETIC   SURVEYING  — 
TRIANGULATION 

1.  Geodesy. 

Geodesy  is  the  science  which  treats  of  investigations  of  the 
form  and  dimensions  of  the  earth's  surface  by  direct  measure- 
ments. The  two  methods  chiefly  employed  in  determining  the 
earth's  figure  are  (i)  the  measurement  of  arcs  on  the  surface, 
combined  with  the  determination  of  the  astronomical  positions 
of  points  on  these  arcs,  and  (2)  direct  observation  of  the  variation 
in  the  force  of  gravity  in  different  parts  of  the  earth's  surface. 

2.  Geodetic  Surveying. 

Geodetic  Surveying  is  that  branch  of  the  art  of  surveying 
which  deals  with  such  great  areas  that  it  becomes  necessary  to 
make  systematic  allowance  for  the  effect  of  the  earth's  curvature. 
In  making  an  accurate  survey  of  a  whole  country,  for  example, 
the  methods  of  plane  surveying  no  longer  suffice,  and  the  whole 
theory  of  locating  points  and  calculating  their  positions  must  be 
modified  accordingly.  Such  surveys  require  the  accurate  loca- 
tion of  points  separated  by  long  distances,  to  control  the  accuracy 
of  subsequent  surveys  for  details,  such  as  coast  charts  and  topo- 
graphic maps,  or  for  national  and  state  boundaries.  The  general 
method  employed  is  that  of  triangulation,  in  which  the  location 
of  points  is  made  to  depend  upon  the  measurement  of  horizontal 
angles,  the  distances  being  calculated  by  trigonometry  instead 
of  being  measured  directly.  This  method  was  first  applied  to 
the  measurement  of  arcs  on  the  earth's  surface  by  Snellius  of 
Holland  in  1615. 


:  GEODETIC  SURVEYING— TRIANGULATION 

Although  we  may  make  this  distinction  when  denning  the 
terms  it  is  not  necessary  to  separate  the  two  in  practice.  It  is 
evident  that  geodetic  surveys  must  be  made  before  accurate 
dimensions  of  the  earth  can  be  computed;  and,  conversely, 
it  is  true  that  before  geodetic  surveys  can  be  calculated  exactly, 
the  earth's  dimensions  must  be  known.  Hence  geodetic  surveys 
are  usually  conducted  with  a  twofold  purpose:  (i)  for  collecting 
the  scientific  data  of  geodesy,  and  (2)  for  mapping  large  areas, 
every  survey  depending  upon  data  previously  determined,  but 
also  adding  to  or  improving  the  data  already  existing.  For  this 
reason  the  measurements  are  made  with  greater  refinement  than 
would  be  necessary  for  practical  purposes  alone. 

3.  Triangulation. 

A  triangulation  system  consists  of  a  network  of  triangles  the 
vertices  of  which  are  marked  points  on  the  earth's  surface.  It  is 
essential  that  the  length  of  one  side  of  some  triangle  should  be 
measured,  and  also  that  a  sufficient  number  of  angles  should  be 
measured  to  make  possible  the  calculation  of  all  the  remaining 
triangle  sides.  In  addition  to  the  measurements  that  are  abso- 
lutely necessary  for  making  these  calculations  it  is  important  to 
have  other  measurements  for  the  purpose  of  verifying  the  ac- 
curacy of  both  the  calculations  and  the  field-work. 

4.  Classes  of  Triangulation. 

Triangulation  is  divided,  somewhat  arbitrarily,  into  three 
grades,  called  primary,  secondary,  and  tertiary,  the  classification 
depending  upon  the  purpose  for  which  the  triangulation  is  to  be 
used  and  upon  the  degree  of  accuracy  demanded.  The  primary 
system  is  planned  and  executed  for  the  purpose  of  furnishing  a 
few  well-determined  positions  for  controlling  the  accuracy  of  all 
dependent  surveys.  Since  the  primary  is  usually  the  only  tri- 
angulation which  is  employed  in  the  purely  scientific  problems 
of  geodesy,  the  selection  of  the  primary  points  will  be  governed 
in  part  by  the  requirements  of  any  geodetic  problem  that  it  is 
proposed  to  investigate.  The  secondary  triangulation  is  some- 
what less  accurate  than  the  primary,  and  the  lines  are  generally 


LENGTH  OF  LINE  3 

shorter;  it  is  often  simply  a  means  of  connecting  the  primary  with 
the  tertiary  system.  Sometimes  the  secondary  is  extended  into 
a  region  which  is  to  be  surveyed  but  which  is  not  covered  at  all 
by  the  primary  triangulation,  and  then  it  becomes  the  con- 
trolling triangulation  of  the  region.  The  tertiary  triangulation 
furnishes  points  needed  for  filling  in  details  on  the  hydrographic 
or  topographic  maps.  It  is  of  a  low  order  of  accuracy  as  com- 
pared with  the  primary,  but  is  amply  accurate  for  controlling 
the  surveys  for  detail.  These  tertiary,  stations  furnish  the  start- 
ing points  for  plane-table  surveys,  traverse  lines,  etc.  All  three 
classes  of  triangulation  are  not  necessarily  present  in  a  survey 
unless  it  be  a  very  extensive  one.  In  surveys  of  minor  import- 
ance there  may  be  but  one  class  of  triangulation. 

5.  Length  of  Line. 

The  length  of  line  which  may  be  used  is  determined  largely  by 
the  character  of  the  country  to  be  surveyed.  In  California, 
where  the  mountains  are  high  and  the  atmosphere  is  exception- 
ally clear,  the  network  of  triangulation  known  as  the  "  Davidson 
quadrilaterals"  (Fig.  i)  is  composed  of  lines  varying  in  length 
from  50  to  over  150  miles;  whereas  in  flat  country,  lines  from 
15  to  25  miles  long  are  the  most  common.  Although  the 
progress  of  the  triangulation  is  apparently  more  rapid  when 
long  lines  are  used,  it  is  not  necessarily  economical  to  use  very 
long  sights.  The  time  gained  by  having  but  few  stations  to 
occupy  may  be  more  than  offset  by  the  delays  due  to  unfavorable 
atmospheric  conditions.  Furthermore,  it  may  be  necessary  to 
introduce  many  additional  stations  in  the  detail  surveys  in  order 
to  reach  all  parts  of  the  area  to  be  mapped.  The  accuracy  of 
triangulation  is  not  appreciably  lessened  by  using  rather  short 
lines.  In  planning  the  system  an  attempt  should  be  made  to  use 
that  length  of  line  which  will  result  in  the  greatest  economy, 
taking  into  consideration  the  cost  of  reconnoissance,  signal 
building,  base-line  measurement,  and  the  measurement  of  the 
angles. 


4   GEODESY  AND  GEODETIC  SURVEYING  —  TRIANGULATION 

6.   Check  Bases. 

It  has  already  been  stated  that  at  least  one  line  in  a  system 
must  be  measured.  In  order  to  verify  the  accuracy  of  all  the 
measurements,  it  is  customary  to  introduce  additional  base  lines 
into  the  triangulation  at  intervals  varying  from  50  to  500  miles. 


Mt.Shasta 


SCALE  OF   MTLE8 


0          20         10          60          80        100 

FIG.  i.    Primary  Triangulation  in  California  (Davidson  Quadrilaterals). 

The  lengths  of  these  bases  may  be  found  by  calculation  of  the  tri- 
angles as  well  as  by  the  direct  measurement;  this  furnishes  a 
most  valuable  check  on  the  accuracy  of  the  field  work.  In  the 
triangulation  of  the  United  States  Coast  and  Geodetic  Survey 
the  frequency  with  which  these  check  bases  should  occur  is  de- 


CHECK  BASES 


termined  by  the  strength  of  the  chain  of  triangulation  as  found 
by  the  method  given  in  Art.  8.     The  factor  RI  (Equa.  [a])  be- 
tween bases  should  be  about  130  for  primary  work,  although  this 
may  be  increased  to  200  if  necessary. 
In  the  triangulation  of  New  England  there  are  three  bases:  (i) 

Cooper 


Mt.Blue, 


Nanticket 


West 
Hill 


FIG.  2.     Primary  Triangulation  of  New  England. 

the  Fire  Island  base,  about  9  miles  long,  in  the  southern  part  of 
Long  Island;  (2)  the  Massachusetts  base,  about  10  miles  long, 
near  the  Northeast  corner  of  Rhode  Island;  and  (3)  the  Epping 
base,  about  5  miles  long,  in  Maine.  These  base  lines  are  shown 
as  heavy  lines  in  Fig.  2.  The  total  length  of  the  triangulation 
between  the  Epping  and  Fire  Island  bases  is  about  350  miles. 


6      GEODESY  AND   GEODETIC  SURVEYING— TRIANGULATION 

The  accuracy  with  which  the  triangulation  was  executed  is  indi- 
cated by  a  comparison  of  the  measured  and  computed  lengths. 
The  length  of  the  Epping  base  as  calculated  from  the  Fire  Island 
base  is  0.042  meter  less  than  the  measured  length;  the  length  of 
Epping  base  calculated  from  the  Massachusetts  base  is  0.136 
meter  less  than  the  measured  length. 

7.  Geometric  Figure. 

The  geometric  figure  generally  recognized  as  the  best  one  for 
triangulation  purposes  is  the  quadrilateral,  consisting  of  four 
stations  joined  by  six  lines,  thus  forming  four  triangles  in  which 
there  are  altogether  eight  independent  angles  to  be  measured. 
This  figure  furnishes  a  greater  number  of  checks  than  any  of  the 
simple  figures  and  therefore  gives  a  good  determination  of  length. 
The  polygon  having  an  interior  station  is  also  a  strong  figure. 
Figures  which  are  more  complex  than  these  usually  make  the 
calculation  troublesome  and  expensive,  while  simpler  figures, 
like  single  triangles,  result  in  diminished  accuracy.  In  the  work 
of  the  United  States  Coast  Survey  the  primary  triangulation  is 
made  up  chiefly  of  complete  quadrilaterals  and  partly  of  polygons 
having  an  interior  station.  In  these  figures  all  of  the  stations 
are  supposed  to  be  occupied  with  the  triangulation  instrument, 
but  for  secondary  and  tertiary  triangulation  some  stations  may 
be  left  unoccupied. 

8.  Strength  of  Figure. 

In  deciding  which  of  several  possible  triangulation  schemes 
should  be  adopted  it  is  essential  to  inspect  the  chain  of  triangles 
with  a  view  to  ascertaining  which  is  the  strongest  geometric 
figure,  that  is,  which  one  will  give  the  calculated  length  of  the 
final  line  with  the  least  error  due  to  the  shape  of  the  triangles. 

An  estimate  of  the  uncertainty  in  the  computed  side  of  a  tri- 
angle is  given  by  its  probable  error  as  found  by  the  method  of  least 
squares.  The  square  of  the  probable  error  (p)  of  a  triangle  side 
as  computed  through  a  chain  of  triangles  is  given  by  the  equation 

f  =  3 


STRENGTH  OF  FIGURE  7 

< 

in  which  d  is  the  probable  error  of  an  observed  direction,  Nd  is  the 
number  of  directions  observed,  Ne  is*  the  number  of  geometric 
conditions  that  must  be  satisfied  in  the  figure,  and  8A  and  dB  are 
the  differences  in  the  log  sines  corresponding  to  a  difference  of  i" 
in  the  angles  A  and  B,  A  being  opposite  the  known  side  and  B 
opposite  the  computed  side.  A  and  B  are  known  as  the  distance 
angles.  The  2}  indicates  that  the  quantity  in  brackets  is  to  be 
computed  for  each  triangle  in  the  chain  and  the  sum  of  these 

numbers  used  in  the  formula.     The  factor  —  ^—  —  -  depends  upon 


the  kind  of  figure  chosen  and  the  factor  2  [&AZ  +  &A&B  +  5B2] 
depends  upon  the  shape  of  the  triangles  of  which  the  figure  is 
composed;  hence  the  product  of  the  two  is  a  measure  of  the 
strength  of  figure  and  is  independent  of  the  precision  with  which 
the  angles  themselves  are  measured.  The  strength  R  of  any 
figure  is  therefore  given  by  the  equation 

*  =  ^ 


The  smaller  the  value  of  this  product  the  more  favorable  the 
geometric  conditions,  and  the  stronger  the  figure. 

If  the  value  of  this  product  be  computed  for  every  possible 
route  through  the  triangulation  system,  there  will  result  a  mini- 
mum value  (RI)  for  the  best  chain  of  triangles,  a  second  best 
value  C#2),  and  a  third  and  fourth,  and  so  on.  It  will  be  found 
that  the  chain  of  triangles  having  the  greatest  influence  in  fixing 
the  length  of  the  final  line  is  that  corresponding  to  RI,  or  the  best 
chain.  The  second-best  chain  will  have  some  influence,  and  the 
third  and  fourth  correspondingly  less.  Hence,  in  choosing  be- 
tween two  or  more  possible  systems  of  triangulation  which  join  a 
given  base  with  some  specified  line,  that  route  having  the  smallest 
RI  is  to  be  preferred,  unless  RI  proves  to  be  nearly  the  same  for 
the  different  routes,  in  which  case  that  chain  having  the  smallest 
Rz  would  be  chosen. 

As  an  example  of  the  way  in  which  the  preceding  method  would 


8   GEODESY  AND  GEODETIC  SURVEYING  — -  TRIANGULATION 

be  applied,  take  the  case  of  the  quadrilateral  shown  in  Fig.  3. 
Assuming  the  base  AB  to  be  already  fixed  in  direction,  the  point 
C  is  then  determined  by  observing  the  new  directions  AC  and 
BC.  D  is  fixed  by  the  directions  AD  and  BD.  In  addition  to 
these  four  the  directions  CB,  CA,  CD,  DC,  DB,  DA  are  all  ob- 
served. This  gives  10  observed  directions  as  the  value  of  N& 


FIG.  3. 

In  determining  the  number  of  geometric  conditions  it  is  seen  that 
there  are  four  triangles,  and  that  in  each  triangle  the  sum  of  the 
three  angles  must  equal  a  fixed  amount,  180°  +  the  spherical 
excess  of  that  triangle.  It  will  be  found,  however,  that  if  any 
three  of  these  triangles  are  made  to  fulfill  these  conditions,  the 
fourth  will  necessarily  do  so,  and  hence  is  not  really  independent; 
in  other  words,  there  are  but  three  conditions  dependent  upon 
the  closure  of  the  triangles.  In  addition  to  these  three  angle 
conditions  there  is  also  a  distance  check;  that  is,  the  angles  must 
be  so  related  that  the  computed  length  of  side  CD  is  the  same,  no 
matter  which  pair  of  triangles  is  used  in  making  the  computation. 
The  angles  of  the  triangle  may  in  each  case  add  up  to  the  correct 
amount,  and  yet  the  figure  will  not  be  a  perfect  quadrilateral 
unless  this  last  condition  is  fulfilled.  There  are  then,  in  all,  four 
geometric  conditions  existing  among  the  angles  (Nc  =  4). 
Therefore  the  factor  for  the  completed  quadrilateral  is 
Nd  -  Nc  _  joj^j.  = 
Nd  io  =  ° 


NUMBER  OF  CONDITIONS  IN  A  FIGURE  9 

In  the  triangle  ADB  the  distance  angles  for  computing  the 
diagonal  are  DAB  and  ADB,  that  is,  71°  and  71°.  The  difference 
for  i"  for  71°  is  0.72  in  units  of  the  6th  decimal  place.  The 
quantity  in  brackets  in  the  formula  is  therefore  (0.52  +0.52  + 
0.52)  =  1.56,  or  2  to  the  nearest  unit.  In  Table  I  these  numbers 
are  given  for  all  combinations  of  angles  which  will  occur  in  prac- 
tice, so  that  this  factor  may  be  found  at  once  by  entering  the  table 
with  the  two  distance  angles.  For  the  triangle  BDC  the  distance 
angles  for  computing  the  side  DC  are  93°.  and  38°,  the  tabular 
number  being  7.  For  this  chain  of  triangles,  then,  RI  =  0.6  X 
(2  -f  7)  =  5.4.  For  triangle  BA C  the  angles  are  76°  and  62°, 
and  the  number  equals  2.  For  triangle  DC  A  the  angles  are  120° 
and  29°,  and  the  number  equals  n.  Therefore  R2  =  0.6  X  13  = 
7.8.  If  we  compute  CD  through  the  triangles  ACB  and  DCB, 
we  find  RZ  =  15.6.  Using  triangles  DBA  and  DC  A,  Ri  =  30.6. 
In  comparing  the  strength  of  this  quadrilateral  with  that  of  any 
other  figure,  reliance  would  be  placed  mainly  upon  RI  =  5. 4  and 
partly  upon  R2  =  7.8. 

Following  are  the  values  of  factor  -  d        -  for  several  figures 

lid  • 

frequently  used  in  triangulation: —  single  triangle,  0.75;  quad- 
rilateral, 0.60;  quadrilateral  with  one  station  on  fixed  line 
not  occupied,  0.75;  quadrilateral  with  one  station  not  on  fixed 
line  not  occupied,  0.71;  triangle  with  interior  station,  0.60;  tri- 
angle with  interior  station,  one  station  on  fixed  line  not  occupied, 
0.75;  triangle  with  interior  station,  one  station  not  on  fixed  line 
not  occupied,  0.71;  four-sided  figure  with  interior  station,  0.64; 
five-sided  figure  with  interior  station,  0.67;  six-sided  figure  with 
interior  station,  0.68.  (For  additional  cases  see  General  In- 
structions for  the  Field  Work  of  the  Coast  and  Geodetic  Survey, 
1908;  or  Special  Publication  No.  26.) 

9.  Number  of  Conditions  in  a  Figure. 

In  determining  the  number  of  conditions  in  any  figure  it  is  well 
to  proceed  by  plotting  the  figure  point  by  point,  and  to  write 
down  the  conditions  as  they  arise,  but  it  will  be  of  assistance  to 


10     GEODESY  AND   GEODETIC   SURVEYING  —  TRIANGULATION 

have  a  check  on  the  results  obtained  by  this  process.  If  n  rep- 
resents the  total  number  of  angles  measured,  and  5  the  number 
of  stations,  then,  since  it  requires  two  angles  to  locate  a  third 
point  from  the  base  line,  two  more  to  locate  a  fourth  point  from 
any  two  of  these  three  points,  and  so  on,  the  number  of  angles 
required  is  2  (s  —  2);  and  since  each  additional  angle  gives  rise 
to  a  condition,  the  number  of  conditions  will  equal  the  number 
of  superfluous  angles,  or 

Nc  =  n  —  2  (s  —  2) 
=  n  —  25  +  4. 

For  example,  in  a  quadrilateral  in  which  one  station  is  unoccupied 
there  are  six  angles  measured,  and  #,.  =  6  —  8+4  =  2. 
The  number  of  conditions  may  also  be  found  from  the  equation 

Nc  =  2  /  -  k  ~  3  s  +  Su  +  4, 

where  /   =  the  total  number  of  lines, 

/i  =  the  number  of  lines  sighted  in  one  direction  only, 

s  =  the  total  number  of  stations, 
and      su  =  the  number  of  unoccupied  stations. 

In  the  preceding  example  this  equation  becomes 
#0  =  12-3-12  +  1+4  =  2. 

10.  Allowable  limits  of  R^  and  RQ. 

In  the  primary  triangulation  of  the  United  States  Coast  and 
Geodetic  Survey,  the  extreme  limits  for  R±  and  R^  between  base 
nets  are  25  and  80,  respectively.  These  are  reduced  to  15  and  20 
if  this  does  not  increase  the  cost  over  25  per  cent.  For  secondary 
triangulation  the  limits  for  RI  and  R%  are  50  and  150;  these  are 
reduced  to  25  and  80  if  the  cost  is  not  more  than  25  per  cent 
greater.  For  tertiary  triangulation  the  50  and  150  limit  may  be 
exceeded  if  it  appears  necessary.  As  stated  in  Art.  6,  when  Rt 
has  accumulated  to  130  between  bases,  a  new  base  line  should 
be  introduced  as  a  check  on  the  accuracy  of  the  calculated  lengths. 
If  the  character  of  the  country  is  such  that  a  base  cannot  be 
located  at  this  point,  RI  may  be  increased  to  200  if  necessary. 


CALCULATION  OF  HEIGHT  OF  OBSERVING  TOWER          II 

1 1 .  Reconnoissance. 

The  work  of  planning  the  system  is  in  many  respects  the  most 
important  part  of  the  project  and  demands  much  experience  and 
skill.  Upon  the  proper  selection  of  stations  will  depend  very 
largely  the  accuracy  of  the  result,  as  well  as  the  cost  of  the  work. 
No  amount  of  care  in  the  subsequent  field-work  will  fully  com- 
pensate for  the  adoption  of  an  inferior  scheme  of  triangulation. 
Three  points  in  particular  will  have  to  be  kept  in  mind  in  planning 
a  survey:  (i)  the  "strength"  of  the  figures  adopted;  (2)  the  dis- 
tribution of  the  points  with  reference  to  the  requirements  of  the 
subsequent  detail  surveys;  and  (3)  the  cost  of  the  work.  In  de- 
ciding which  stations  to  adopt  it  is  desirable  to  make  a  prelimi- 
nary examination  of  all  available  data,  such  as  maps  and  known 
elevations.  If  no  map  of  the  region  exists,  a  sketch  map  must  be 
made  as  the  reconnoissance  proceeds.  While  much  information 
may  be  obtained  from  such  maps  as  are  available,  the  final  de- 
cision regarding  the  adoption  of  points  must  rest  upon  an  exami- 
nation made  in  the  field.  All  lines  should  be  tested  to  see  if  the 
two  stations  are  intervisible.  This  may  be  done  by  means  of 
field  glasses  and  heliotrope  signals.  In  cases  where  the  points 
are  not  intervisible,  owing  to 'intervening  hills  or  to  the  curvature 
of  the  earth's  surface,  it  will  be  necessary  to  determine  approxi- 
mately, by  means  of  vertical  angles  or  by  the  barometer,  the 
elevation  of  the  proposed  stations  and  of  as  many  intermediate 
points  as  may  be  required,  and  then  to  calculate  the  height  to 
which  towers  will  have  to  be  built  in  order  to  render  the  proposed 
stations  visible.  If  the  height  of  the  towers  is  such  as  to  make 
the  cost  prohibitive,  the  line  must  be  abandoned  and  another 
scheme  of  triangles  substituted. 

12.  Calculation  of  Height  of  Observing  Tower. 

After  determining  the  elevations  of  the  stations  and  the  inter- 
vening hills  along  a  line,  as  well  as  the  distances  between  them, 
the  height  of  the  tower  required  may  be  found  by  the  following 
method:  The  curvature  of  the  earth's  surface  causes  all  points 
to  appear  lower  than  they  actually  are.  A  hill  appearing  to  be 


12     GEODESY  AND    GEODETIC  SURVEYING— TRIANGULATION 


exactly  on  the  level  of  the  observer's  eye  is  in  reality  higher  above 
sea-level  than  the  observer.  The  light  coming  from  the  hill  to 
the  observer's  eye  does  not,  however,  travel  in  a  straight  line, 
but  is  bent,  or  refracted,  by  the  atmosphere  into  a  curve  which 
is  concave  downward  and  is  approximately  circular.  The  result 
is  that  the  object  appears  higher  than  it  would  if  there  were  n@ 
refraction.  The  amount  of  the  apparent  change  in  height  due 
to  refraction  is  found  to  be  only  about  one-seventh  part  of  the 
apparent  depression  due  to  curvature.  Since  these  two  correc- 
tions always  have  opposite  signs  and  have  a  nearly  fixed  relation 

to  each  other,  it  is  sufficient  in  prac- 
tice to  calculate  the  correction  to  the 
difference  in  height  due  to  both  cur- 
vature and  refraction,  and  to  treat 
the  combined  correction  as  though  it 
were  due  to  curvature  alone,  since  the 
curvature  correction,  being  the  larger, 
always  determines  which  way  the  total 
correction  shall  be  applied. 

In  Fig.  4,  A  is  the  position  of  the 
observer,'  looking  in  a  horizontal  di- 
rection toward  point  B.    BC  is  the  amount  by  which  B  appears 
lower  than  it  really  is,  since  A  and  C  are  both  at  the  same  eleva- 
tion (sea-level). 
By  geometry,  BC  :  AB  =  AB  :  BD 

ft 

or  BC  =  —  • 

BD 

Since  BC  is  small  compared  with  BD,  the  percentage  error  is 
small  if  we  call  AB  =  AC  and  BD  =  the  diameter  of  the  earth, 
whence 

(dist.)2 


FIG.  4. 


BC 


diameter 


(approx.). 


The  light  from  Bf  (Fig.  5)  follows  the  dotted  curved  path  which 
is  tangent  to  the  sight  line  at  A.    The  observer  therefore  sees  Bf 


CALCULATION  OF  HEIGHT  OF  OBSERVING  TOWER 


at  B.     In  order  to  find  the  relation  of  BB'  to  BC  it  is  convenient 
to  employ  w,  the  coefficient  of  refraction,  which  is  denned  as  the 
number  by  which   the    central  angle  AOB 
must  be  multiplied  in  order  to  obtain  the 
angle  BAB'-,  therefore 

angle  of  refraction  =  2  X  m  X  BAC. 

Since  these  angles  are  small,  distances  BB' 
and  BC  are  nearly  proportional  to  the  angles 
themselves,  hence 

BB' :  BC  =  BAB' :  BAC 
and  BB'  =  2  m  X  BC. 


The  net  correction  (B'C  =  h)  is  the  difference 
between  the  two,  that  is 

h  =  BC  -  BB' 
__(dist.)2 
diam. 
_  (dist.)2 
diam. 


FIG.  5. 


—  2  m 


(dist.)2 
diam. 

(i  -  2m). 


The  mean  value  of  m  is  found  to  be  about  0.070.  Substituting 
this,  and  the  value  for  the  earth's  diameter,  and  reducing  h  to 
feet,  we  have 

h  (in  feet)  =  K2  (in  miles)  X  0.574, 
or  K  (in  miles)  =  Vh  (in  ft.)  X  1.32, 

in  which  K  is  the  distance  in  miles.  Values  of  h  and  K  for 
distances  up  to  60  miles  will  be  found  in  Table  II. 

As  an  example  of  how  this  formula  is  applied,  suppose  it  is  de- 
sired to  sight  from  A  to  B  (Fig.  6),  and  that  a  hill  C  obstructs 
the  line.  At  A  draw  a  horizontal  line  AD  and  also  a  curve  AE 
parallel  to  sea-level.  The  distance  from  the  tangent  to  the  dotted 

K2 
curve  at  C  is  — ,  which  for  46  miles  is  1411.9  ft.     Similarly, 

K2- 
at  B,  — =  4708.0  ft.    But  since  the  ray  of  light  from  B  to  A 


14     GEODESY  AND   GEODETIC  SURVEYING  —  TRIANGULATION 

is  curved,  B  is  seen  at  Bf,  or  659.2  ft.  nearer  to  the  tangent  AD\ 
similarly,  C  appears  to  be  197.7  ft.  nearer  the  tangent  line. 
Therefore,  in  deciding  the  question  of  visibility  we  may  compute 
the  combined  correction  and  say  at  once  that  the  curve  at  C  is 


D 


~38, 


FIG.  6. 


1214.2  ft.  below  AD,  and  at  B  is  4048.8  ft.*  below  AD.  Adding 
2300  ft.  (the  elevation  of  A)  to  each  of  these  values  of  h,  we  obtain 
the  (vertical)  distances  from  the  tangent  line  down  to  sea-level, 
namely  3514.2  ft.  and  6348.8  ft.  at  C  and  B}  respectively.  Sub- 


84  miles 


B' 


FIG.  7. 


tracting  the  elevations  of  C  and  B,  we  obtain  2464.2  ft.  and  4548.8 
ft.  as  the  distances  of  points  C  and  D  below  the  tangent  line  AD. 
The  three  points  are  now  referred  to  a  straight  line  (the  tangent), 
and  the  question  of  visibility  is  determined  at  once  by  similar 

*  Since  the  table  extends  only  to  60  miles,  the  value  of  h  is  first  found  for  half 
the  distance  (42  mi.),  and  the  result  multiplied  by  4. 


CALCULATION  OF   HEIGHT   OF   OBSERVING  TOWER          15 

triangles.  In  Fig.  7  it  will  be  seen  that  the  straight  line  from  Bf 
to  A  is  ||  X  4548.8  =  2491.0  ft.  below  the  tangent  (opposite  C), 
and  consequently  is  26.8  ft.  lower  than  C.  Twenty-seven-foot 
towers  would  therefore  barely  make  B'  visible  from  A .  In  order 
to  avoid  the  atmospheric  disturbances  near  the  ground  at  C  the* 
towers  would  really  have  to  be  carried  up  to  a  height  of  40  ft.  or 
even  more.  Of  course  the  line  of  sight  is  not  actually  straight 
between  A  and  B,  as  shown  in  the  diagram;  but  this  method  of 
solving  the  problem  gives  the  same  result  as  though  the  curva- 
ture and  refraction  were  dealt  with  separately  and  the  sight  lines 
all  drawn  curved. 

If  it  were  required  to  find  the  heights  of  towers  necessary  to 
make  it  possible  to  sight  from  A  across  a  water  surface  to  Z>,  we 
should  proceed  as  follows:  Suppose  the  elevation  of  A  above  the 
water  surface  is  20  ft.  and  that  of  D  is  10  ft.  From  A  we  may 
draw  a  line  tangent  to  the  water-level  at  T  (Fig.  8).  Knowing 


FIG.  8. 


the  height  of  A,  we  may  find  the  distance  AT  from  Table  II. 
Subtracting  this  distance  from  AD,  we  find  the  distance  TD. 
From  this  latter  distance  we  may  compute  the  height  of  the  tan- 
gent line  above  the  surface  at  D,  and,  finally,  knowing  the  height 
of  Z>,  we  find  the  distance  of  D  below  the  tangent  line.  Now 
that  the  points  are  referred  to  a  straight  line,  we  have  at  once 
the  height  of  tower  required  on  D  alone.  If  the  two  towers  are 
to  be  of  equal  height,  we  may  estimate  the  required  height 
closely  and  then  verify  the  result  by  a  second  computation,  add- 
ing the  assumed  height  of  the  tower  to  the  elevation  of  A . 

If  it  is  desired  to  keep  the  line  of  sight  at  least  10  ft.  above  the 
surface  at  every  point  in  order  to  avoid  errors  due  to  excessive 
refraction,  we  may  draw  a  parallel  curve  10  ft.  above  the  water 
surface  and  solve  the  problem  as  before.  The  difference  in  radii 


16     GEODESY  AND   GEODETIC  SURVEYING— TRIANGULATION 

of  the  two  curves  will  not  have  an  appreciable  effect  on  the  com- 
puted values  of  h  and  K. 

13.   Method  of  Marking  Stations. 

The  importance  of  permanently  marking  a  trigonometric  sta- 
tion and  connecting  it  with  other  reference  marks  cannot  be 
easily  overestimated,  since  by  this  means  we  may  avoid  the  costly 
work  of  reproducing  triangulation  points  which  have  been  lost. 

When  the  station  is  on  ledge,  the  point  is  best  marked  by 
making  a  fairly  deep  drill  hole  and  setting  a  copper  bolt  into  it. 
A  triangle  is  chiseled  around  the  hole  as  an  aid  in  identifying  the 
point.  Other  drill  and  chisel  marks  should  be  made  in  the 
vicinity,  and  their  distances  and  directions  from  the  center  mark 
determined;  these  will  serve  as  an  aid  in  recovering  the  position 
of  the  center  mark-  in  case  it  is  lost. 

If  the  station  is  on  gravel  or  other  soft  material,  the  station 
mark  on  the  surface  is  usually  a  stone  or  concrete  post,  set  deep 
enough  to  be  unaffected  by  frost  action  and  having  a  drill  hole 
or  other  distinguishing  mark  on  top.  There  is  usually  also  a  sub- 
surface mark,  such  as  a  second  stone  post,  a  bottle  or  a  circular 
piece  of  earthenware,  placed  some  distance  below  the  surface 
mark,  to  preserve  the  location  in  case  the  latter  is  lost.  The 
Coast  and  Geodetic  Survey  and  the  United  States  Geological 
Survey  use  cast  metal  discs  provided  with  a  shaft  ready  to  place 
in  concrete,  and  bearing  an  inscription  giving  the  name  of  the 
organization  and  other  information.  (See  Figs,  ga  and  gb.) 

The  following  description  and  sketch  are  given  to  illustrate  a 
description  of  a  triangulation  station. 

Triangulation  Station  "  Beacon  Rock." 

The  station  is  in  the  town  pf  ,  ,  on  a  hill  on  the 

property  of  John  Smith  situated  on  the  north  side  of  the  road  from  Bourne  to 
Canterbury.  It  is  reached  by  a  trail  which  leaves  the  road  at  a  point  about 
250  meters  west  of  Smith's  house.  It  is  about  225  meters  by  trail  to  the 
station.  The  point  is  marked  by  a  one-inch  copper  bolt  set  in  a  drill  hole  in 
the  ledge  and  with  a  triangle  chiseled  around  it,  and  by  witness  marks  as  shown 
in  the  accompanying  sketch.  The  hill  is  somewhat  wooded  to  the  north  and 
west,  but  there  is  a  clear  view  in  all  other  directions. 


FIG.  ga.     Triangulation  Station  Mark. 
(Coast  and  Geodetic  Survey.) 


FIG.  gb.    Reference  Mark. 
(Coast  and  Geodetic  Survey.) 


1 8     GEODESY   AND    GEODETIC   SURVEYING  —  TRIANGULATION 

DISTANCES  AND   AZIMUTHS   FROM   CENTER 

Station.  Azimuth.  Dist.  to  drill  hole. 

71. 3m. 
41.0  m. 
21.47  m. 
101.  2  m. 
78.3401. 


Holder 

21°  50' 

Bear  Hill  

121°  16' 

Witness  Mark  

Dayton   . 

o          / 

Witness  Mark  

283°   05' 

Sheen  Id.  .  . 

^2<°    A.O' 

FIG.  10.     Sketch  of  Triangulation  Station. 

14.   Signals,  Tripods. 

In  order  that  the  exact  position  of  the  station  may  be  visible 
to  the  observer  when  measuring  the  angles,  a  signal  of  some  sort 
is  erected  over  the  station.  For  comparatively  short  lines,  less 
than  about  15  miles,  the  tripod  signal  is  often  sufficient.  (See 
Fig.  u.)  It  is  not  expensive  to  build,  saves  the  cost  of  a  man  to 


HELIOTROPES  19 

attend  signal  lights  (as  is  necessary  with  heliotropes  or  acetylene 
lights),  and  permits  setting  the  instrument  over  the  station  with- 
out removing  the  signal.  It  usually  consists  of  a  mast  of  4"  x  4" 
spruce,  with  legs  of  about  the  same  size.  Three  horizontal  braces 
of  smaller  dimensions  (2"  X  3")  tie  the  mast  to  the  legs,  and 
three  longer  horizontal  braces  are  nailed  to  the  legs.  If  the 


FIG.  ii.    Tripod  Signal. 

signal  is  very  large,  additional  sets  of  braces  may  be  put  on,  to 
give  greater  stiffness.  The  size  of  the  mast  may  be  increased  by 
nailing  on  one-inch  boards,  giving  a  mast  6"  X  6". 

15.  Heliotropes. 

When  sighting  over  longer  lines  it  is  necessary  to  use  heliotrope 
signals  if  observing  by  day,  and  acetylene  lights  if  observing  by 
night.  The  heliotrope  is  simply  a  plane  mirror  with  some  device 
for  pointing  it  so  that  reflected  sunlight  will  reach  the  distant 


20     GEODESY  AND   GEODETIC   SURVEYING  —  TRIANGULATION 

station.  The  two  more  common  heliotropes  are  (i)  the  one  in 
which  the  light  is  directed  through  two  circular  rings  of  slightly 
different  diameters  (Fig.  12),  and  (2)  that  known  as  the  Steinheil 
heliotrope  (Fig.  13). 

The  ring  heliotrope  consists  essentially  of  two  circular  metal 
rings,  of  slightly  different  diameters,  mounted  on  a  frame,  and  a 
mirror  mounted  in  line  with  the  two  rings  in  such  a  manner  that 
it  can  be  moved  about  two  axes  at  right  angles  to  each  other. 
For  convenience  in  observing  distant  stations  these  two  rings  and 
the  mirror  are  often  mounted  on  the  barrel  of  a  telescope.  The 


FIG.  12.    Heliotrope. 

rings  should  be  so  mounted  that  the  line  between  the  centers  of 
the  rings  may  be  adjusted  parallel  to  the  line  of  sight  of  the  tele- 
scope. In  using  the  heliotrope  the  axis  of  the  rings  is  pointed  by 
means  of  threads  which  mark  the  center  of  the  openings,  or  by 
means  of  the  telescope  itself  after  the  axis  of  rings  and  the  line  of 
sight  of  the  telescope  have  been  made  parallel.  Since  the  sun's 
apparent  diameter  is  about  o°  32',  the  angle  of  the  cone  of  rays 
reflected  from  the  mirror  is  also  o°  32'.  It  is  not  necessary, 
therefore,  to  point  the  beam  of  light  with  great  precision.  If  the 
central  ray  is  nearly  a  quarter  of  a  degree  to  one  side  of  the 
station,  there  will  still  be  some  light  visible  to  the  observer  at  the 
distant  station.  On  account  of  the  rapidity  of  the  sun's  motion 
it  is  necessary  to  reset  the  heliotrope  mirror  at  intervals  of  about 
one  minute. 


HELIOTROPES 


21 


FIG.  isa.    Steinheil  Heliotrope. 


\LL 


V 


To  Station 


FIG.  i3b. 


22     GEODESY  AND   GEODETIC   SURVEYING  —  TRIANGULATION 

The  Steinheil  heliotrope  consists  of  a  mirror  with  both  faces 
ground  plane  and  parallel  and  so  mounted  that  it  can  be  moved 
about  two  axes  at  right  angles  to  each  other.  One  of  these  axes 
is  coincident  with  that  of  a  cylindrical  tube  which  contains  a  small 
biconvex  lens  and  a  white  surface  (usually  plaster  of  Paris)  for 
reflecting  light.  This  tube  may  be  moved  about  two  other  axes 
at  right  angles  to  each  other.  A  small  circular  portion  of  the 
glass  in  the  center  of  the  mirror  is  left  unsilvered,  so  that  light 
may  pass  through  the  glass  plate  down  into  the  tube. 

In  pointing  the  Steinheil  heliotrope  the  cylindrical  tube  con- 
taining the  lens  must  be  pointed  toward  the  sun,  so  that  the  light 
which  passes  through  the  hole  in  the  mirror  will  pass  through  the 
lens,  and,  after  reflection  from  the  plaster  surface,  will  again  pass 
through  the  lens  to  the  back  surface  of  the  mirror,  there  to  be 
partly  reflected  and  partly  transmitted  through  the  glass. 
Keeping  the  tube  in  this  position,  the  mirror  itself  must  be  so 
turned  that  the  spot  of  light  made  visible  by  this  last  reflection 
will  appear  to  cover  the  hill  or  station  to  which  the  light  is  to  be 
Sent. 

One  form  of  heliotrope,  in  use  by  the  Coast  Survey,  called  a 
box  heliotrope,  consists  of  a  pair  of  rings  with  a  mirror  mounted 
behind  them,  and  with  sights  above  the  rings  for  pointing.  A 
telescope  is  mounted  to  one  side  of  and  parallel  to  the  heliotrope. 
The  various  parts  remain  in  position  in  the  box  when  in  use. 
(Fig.  14.) 

The  size  of  mirror  used  in  any  heliotrope  must  be  regulated 
according  to  the  length  of  line  and  the  atmospheric  conditions. 
Most  heliotropes  are  provided  with  some  arrangement  for  varying 
the  size  of  the  opening  through  which  the  light  passes.  If  the 
exposed  portion  of  the  mirror  subtends  an  angle  of  about  o./72 
the  amount  of  light  will  be  sufficient  for  average  conditions. 
This  is  equivalent  to  making  the  diameter  of  the  opening  about 
one-half  inch  for  each  ten  miles.  Different  atmospheric  condi- 
tions will  require  different  openings. 

All  heliotropes  are  provided  with  a  second  mirror,  usually 


ACETYLENE  LIGHTS  23 

larger  than  the  first,  called  the  back  mirror;  this  is  to  be  used 
whenever  the  angle  between  the  sun  and  the  station  is  too  great 
to  permit  sending  the  ray  by  a  sing  e  reflection.  The  back 
mirror  is  set  so  as  to  throw  light  onto  the  first  mirror  and  the 
heliotrope  is  then  adjusted  to  the  reflection  of  the  sun  as  it 
appears  in  the  back  mirror. 


FIG.  14.    Box  Heliotrope. 

(Coast  and  Geodetic  Survey.) 

1 6.   Acetylene  Lights. 

In  the  triangulation  along  the  ninety-eighth  meridian,  in  1902, 
the  Coast  and  Geodetic  Survey  experimented  with  acetylene 
lights  for  triangulation  at  night.  These  experiments  were  suc- 
cessful, and,  owing  to  the  fact  that  the  work  could  usually  pro- 
ceed regardless  of  clouds,  the  use  of  lights  resulted  in  greater 


24     GEODESY  AND   GEODETIC   SURVEYING — TRIANGULATION 

economy  than  observations  by  daylight.  The  lamps  used  at 
first  were  ordinary  acetylene  bicycle  lamps  remodeled  in  the 
instrument  division  of  the  Survey.  The  front  door  of  the  lamp 
was  removed  and  the  ordinary  lens  replaced  by  a  pair  of  condens- 
ing lenses  5  inches  in  diameter.  When  in  use  the  lamp  is  secured 
to  the  platform  by  means  of  a  screw,  and  may  be  moved  both  in 


FIG.  15.    Acetylene  Signal  Lamp. 
(Coast  and  Geodetic  Survey.) 


altitude  and  in  azimuth.  A  small  tube  is  fastened  to  the  top  of 
the  lamp  for  pointing  it  toward  the  observer's  station.  The 
entire  outfit,  including  a  5-lb.  can  of  carbide,  weighs  but  2i|  Ibs. 
(See  Coast  and  Geodetic  Survey  Report  for  1903,  p.  824.)  The 
recent  practice  of  the  Survey  is  to  use  automobile  lamps  in  place 
of  the  bicycle  lamps.  (Fig.  15.) 


TOWERS  25 

17.  Towers. 

Where  a  line  is  obstructed  by  hills  or  woods,  or  where  the 
curvature  of  the  earth  is  sufficient  to  make  the  station  invisible, 
it  becomes  necessary  to  construct  towers.  If  there  is  much 
heavy  timber  about  the  station,  placing  the  instrument  station 


FIG.  16.    Eighty-foot  Tower. 
(Coast  and  Geodetic  Survey.) 

on  the  ground  may  necessitate  so  much  cutting  that  it  will  be 
more  economical  to  construct  a  tower  than  to  cut  the  timber. 

The  form  of  tower  now  used  by  the  United  States  Coast  Survey 
is  very  light  and  slender  as  compared  with  the  older  ones.  This 
kind  of  tower  (Fig.  16)  admits  of  more  rapid  construction  and 


26     GEODESY  AND   GEODETIC  SURVEYING-—  TRIANGULATION 

can  be  built  at  a  lower  cost;  it  is  sufficiently  rigid  to  withstand 
all  ordinary  storms.  The  manner  of  framing  the  tower  is  shown 
in  the  cut  (Fig.  17).  When  the  ties  are  nailed  on,  the  legs  are 
sprung  slightly  into  the  form  of  a  bow,  thus  giving  additional 
stiffness  to  the  structure. 

One  side  of  the  inner  tripod,  which  is  to  support  the  instrument, 
is  first  framed  on  the  ground.  This  side  and  the  third  leg  of  the 
tripod  are  raised  into  position  by  a  fall  and  tackle  and  a  derrick, 
which  may  be  a  tree  or  a  section  of  one  of  the  legs  of  the  outer 
scaffold.  The  derrick  should  be  at  least  two- thirds  the  height 
of  the  piece  to  be  raised.  After  the  tripod  is  raised  and  all  braces 
nailed  on,  it  is  itself  used  as  a  derrick  for  hoisting  the  two  opposite 
frames  of  the  outer  scaffold  into  position.  The  ties  and  braces 
of  the  other  two  sides  are  then  nailed  in  place.  It  should  be 
observed  that  the  inner  and  outer  structures  are  entirely  separate, 
so  that  the  movement  of  the  observer  on  the  platform  of  the 
scaffold  will  not  disturb  the  instrument.  The  legs  of  the  tripod 
and  the  scaffold  are  anchored  by  nailing  them  to  foot  pieces  set 
underground.  The  outer  tower  is  guyed  with  wire  as  a  protec- 
tion against  collapse  in  high  winds. 

This  kind  of  signal  saves  lumber,  transportation,  and  cost  of 
construction;  it  has  a  small  area  exposed  to  the  action  of  the 
wind;  the  short  ties  have  the  effect  of  reducing  the  vibration  due 
to  wind,  which  is  troublesome  in  large  towers;  the  light  keeper 
is  placed  above  the  observer  (10  ft.  or  so)  and  can  operate  his 
lights  without  interfering  with  the  observations.  Another  ad- 
vantage of  these  towers  is  that  the  amount  of  twisting  due  to  the 
sun's  heating  is  found  to  be  exceedingly  small.  (For  further 
details  consult  Coast  Survey  Report  for  1903,  p.  829.) 

The  United  States  Lake  Survey  now  uses  a  tower  constructed 
entirely  of  gas  pipe,  which  has  proved  to  be  more  economical 
than  timber.  It  is  put  together  in  sections  and  hoisted  as  it  is 
built.  The  upper  part  of  the  structure  is  built  first  and  is  then 
hoisted  from  the  ground  by  means  of  tackles ;  the  next  section  is 
then  added  on,  all  the  work  being  done  from  the  ground.  This 


TOWERS 


***    t  0'  Stand  for  lights 
3'    Top  floor 


SIDE  OF  60  FT.  SCAFFOLD 


SIDE  OF  60  FT.  TRIPOD 


1818 


22190 
PLAN  OP  SCAFFOLD  AND  TPIPOD 


Scale  of  Feet 


II 


20 


FIG.  17.     Framing  plan  of  6o-ft.  Tower. 


28     GEODESY  AND   GEODETIC   SURVEYING — TRIANGULATION 


kind  of  tower  is  easy  to  construct,  and  the  material  is  portable; 
the  area  exposed  to  wind  is  very  small. 

Figs.  1 8  and  19  illustrate  small  towers  built  of  green  poles  cut 
near   the   station.     These  towers   were   erected  to  enable  the 

observer  to  see  over  the  dense  growth 
of  timber.  In  the  tower  shown  in 
Fig.  19  standing  trees,  stripped  of 
their  branch.es,  were  utilized  for  two 
of  the  legs  of  the  outer  scaffold. 
18.  Reconnoissance  for  Base  Line. 
With  the  Invar  tape  apparatus,  to 
be  described  in  Chapter  II,  base  lines 
may  now  be  measured  over  much 
rougher  ground  than  was  formerly 
possible,  when  bar  apparatus  was 
used;  still  it  is  advantageous  to  have 
the  base  line  located  in  as  smooth 
and  level  country  as  possible,  provided 
this  does  not  require  weak  triangula- 
tion  to  connect  the  base  with  the  main 
scheme  of  triangles.  The  network  of 
triangles  required  in  making  this  con- 
nection should  be  selected  with  the  same  care  and  according  to 
the  same  principles  as  was  described  for  primary  triangulation. 
In  some  cases  it  is  found  practicable  to  use  the  side  of  a  primary 
triangle  for  the  base  line.  For  example,  in  the  triangulation 
extending  from  Texas  to  California  the  Stanton  base,  which  is 
one  of  the  primary  lines  (8  miles  in  length),  was  measured  directly 
with  the  tape  apparatus. 


FIG.  1 8.    Twenty-five-foot 
Tower  built  of  Green  Poles. 


RECONNOISSANCE    FOR   BASE   LINE 


FIG.  19.    Forty-foot  Tower  built  on  trees  in  place. 


30     GEODESY  AND   GEODETIC  SURVEYING — TRIANGULATION 


PROBLEMS, 

Problem  i.  What  is  the  strength  of  the  quadrilateral  having  all  the  angles  equal 
to  45°  ?  In  case  one  station  on  the  base  is  not  occupied  with  the  instrument,  what 
is  the  strength  ?  If  one  station  not  on  the  base  is  unoccupied,  what  is  the  strength  ? 

Problem  2.     Compare  the  strength  of  the  three  figures  given  in  Fig.  iga. 


FIG. 


Problem  3.  Three  hills  A,B,  and  C  are  in  a  straight  line.  The  distance  from  A 
to  B  is  10  miles  and  the  distance  from  B  to  C  is  15  miles.  The  elevations  are  A  — 
600  ft.,  B  =  550  ft.,  and  C  =  650  ft.  respectively.  Compute  the  height  of  a  tower 
to  "be  built  on  C  the  top  of  which  will  just  be  visible  from  A. 

Problem  4.  Four  hills  A,  B,  C,  and  D  are  in  a  straight  line.  The  elevations  are 
A  =  810  ft.,  B  =  775  ft.,  C  =  1030  ft,  D  =  1300  ft.  respectively.  The  distances 
of  B,  C,  and  D  from  A  are  8  miles,  28  miles,  and  38  miles.  Find  the  height  of 
towers  on  A  and  D  to  sight  over  B  and  C  with  a  lo-ft.  clearance.  The  two  towers 
are  to  be  of  the  same  height. 

Problem  5.  What  angle  is  subtended  by  a  six-inch  mast  at  a  distance  of  twelve 
miles?  SL"' 

Problem  6.  If  a  fourteen-inch  mirror  is  used  on  a  heliotrope  at  a  distance  of  150 
miles,  what  is  the  apparent  angular  diameter  of  the  light?  oV"J 


CHAPTER  II 
BASE  LINES 

19.  Bar  Apparatus  for  Measuring  Bases. 

In  nearly  all  the  earlier  base-line  measurements  (up  to  about 
1885)  the  apparatus  employed  consisted  of  some  arrangement  of 
metal  bars.  Such  apparatus  was  capable  of  yielding  accurate 
results,  but  was  cumbersome  to  use;  consequently  the  base-line 
work  was  a  comparatively  expensive  part  of  the  survey.  An 
account  of  the  development  of  base-measuring  apparatus  will 
be  found  in  Clarke's  Geodesy  and  in  Jordan's  Vermessungskunde, 
Vol.  Ill;  descriptions  of  numerous  forms  used  in  this  country  will 
be  found  in  the  reports  of  the  superintendent  of  the  Coast  and 
Geodetic  Survey. 

20.  Steel  Tapes. 

Experiments  with  the  use  of  steel  tapes  for  base-line  measure- 
ments were  made  by  Jaderin  at  Stockholm  in  1885,  by  the 
Missouri  River  Commission  in  1886,  and  by  Woodward  on  the 
Coast  and  Geodetic  Survey  base  at  Holton,  Indiana,  in  1891. 
The  use  of  steel  tapes  for  this  purpose  was  attended  with  such 
success  that  for  twenty  years  they  were  very  generally  used,  and 
by  1900  they  had  almost  wholly  superseded  the  bar  apparatus 
in  this  country. 

The  greatest  practical  difficulty  encountered  in  the  use  of  steel 
tapes  for  precise  measurement  is  that  of  determinmg  the  true 
temperature  of  the  steel  when  making  the  measurements  in  sun- 
light. The  air  temperature,  as  indicated  by  ordinary  mercurial 
thermometers,  is  seldom  the  correct  temperature  for  the  tape, 
except  during  rainy  weather  or  at  night.  For  this  reason  it  was 
found  necessary  to  make  all  measurements  of  base-lines  at  night 
in  order  to  secure  the  required  accuracy. 

31 


BASE  LINES 


21.  Invar  Tapes. 

In  1906  the  Coast  Survey  made  a  series  of  tests  on  six  primary 
base-lines,  using  the  ordinary  steel  tapes  and  also  several  new  50- 
meter  tapes  made  of  an  alloy  of  nickel  and  steel  called  invar. 
This  alloy  was  discovered  by  C.  E.  Guillaume,  of  the  Interna- 


FIG.  20.    Invar  Tape  on  Reel. 

tional  Bureau  of  Weights  and  Measures,  Paris.  The  tapes  were 
made  by  J.  H.  Agar-Baugh,  of  London.  The  alloy  mentioned 
has  a  very  low  coefficient  of  expansion,  roughly  one-twenty-fifth 
that  of  steel,*  and  consequently  has  a  great  advantage  over  steel 

*  The  coefficient  of  steel  is  about  o.oooon,  that  of  invar  is  about  0.0000004,  f°r 


ACCURACY   REQUIRED  33 

for  base-line  measurement.  The  metal  is  more  easily  bent  than 
steel,  but  with  proper  care  in  handling  the  tapes,  and  with  the 
use  of  fairly  large  reels,  there  is  little  difficulty  in  making  the 
measurements  and  in  securing  the  required  accuracy.  The  re- 
sults of  these  tests  on  the  invar  tapes  may  be  summed  up  as 
follows: 

Measurements  with  invar  tapes  may  be  made  during  daylight 
with  all  the  accuracy  demanded  in  base-line  work,  whereas 
measurements  with  steel  tapes  must  be  made  at  night  in  order  to 
secure  the  required  accuracy. 

In  working  by  daylight  the  errors  of  observation  are  smaller 
and  the  party  can  make  greater  speed  than  when  working  at 
night. 

On  account  of  the  small  temperature  coefficient  of  the  invar 
tape  any  error  due  to  the  failure  of  the  thermometers  to  indicate 
the  true  temperature  of  the  tape  has  much  less  effect  on  the  com- 
puted length  when  the  measurements  are  made  with  invar  than 
when  they  are  made  with  steel. 

Since  it  is  not  necessary  to  standardize  the  invar  tape  in  the 
field,  as  was  always  done  with  the  steel  tape,  the  cost  of  measure- 
ments made  with  the  invar  is  materially  less  than  that  of  measure- 
ments made  with  steel. 

The  superiority  of  these  tapes  has  been  demonstrated  by  re- 
peated trials,  and  they  are  now  used  almost  exclusively  by  the 
Coast  Survey  in  making  base  measurements. 

22.  Accuracy  Required. 

It  is  found  that  there  is  little,  if  any,  advantage  in  measuring 
a  base-line  with  a  precision  greater  than  one  part  in  500,000, 
since  to  do  this  would  give  the  base-line  a  greater  precision  than 
could  be  maintained  in  the  angle  measurements.  There  is  little 
difficulty,  however,  in  obtaining  a  higher  precision;  the  bases 
measured  by  the  Coast  Survey  in  1906  and  1909  show  a  precision 
of  one  part  in  2,000,000  or  better.  It  is  customary  to  divide 
bases  into  sections  of  about  a  kilometer  in  length,  and  to  measure 
each  section  twice.  If  the  two  results  show  a  discrepancy  greater 


34  BASE  LINES 

than  2owm  VK  (K  being  the  number  of  kilometers  in  the  section), 
the  measurements  are  repeated  until  they  do  agree  within  this 
limit;  if  the  first  two  results  agree  within  this  limit,  no  additional 
measurements  are  taken.  This  procedure  is  consistent  with  the 
requirement  that  the  base  be  measured  with  a  precision  of  at 
least  i  in  500,000,  but  that  no  attempt  be  made  to  increase  the 
precision  much  beyond  this  limit. 

23.   Description  of  Apparatus. 

The  invar  tapes  are  usually  about  53  meters  long,  with  two 
graduations  50  meters  apart.  In  some  tapes  a  length  of  one 
decimeter  at  each  end  of  the  5o-meter  length  is  subdivided  into 
millimeters  for  convenience  in  reading.  Intermediate  points  on 
the  tape,  such  as  the  25  meter  point,  are  marked  by  single  lines. 
The  tape  is  about  \  inch  X  -^  inch  in  cross  section  and  weighs 
about  25  grams  per  meter.  This  metal  is  softer  than  steel  and 
has  to  be  wound  on  a  reel  of  at  least  16  inches  diameter  in  order 
to  avoid  permanent  bends  in  the  tape  and  consequent  changes  in 
length.  (Fig.  20.)  In  use  it  is  supported  at  the  ends  and 
usually  at  one  intermediate  point.  The  tension  is  applied  by 
means  of  a  spring  balance  reading  to  25  grams,  the  tension 
ordinarily  used  being  15  kilograms.  An  apparatus  used  for 
applying  the  tension  and  similar  to  that  used  by  the  Coast  Survey 
is  shown  in  Fig.  21.  The  point  of  the  iron  bar  holding  the  spring 
balance  is  pushed  into  the  ground,  and  the  upper  end  is  moved 
right  or  left  to  align  the  tape.  The  adjustable  clamp  makes  it 
possible  to  raise  or  lower  the  balance  so  as  to  bring  the  end  of  the 
tape  to  the  right  grade.  The  spring  balance  employed  is  a  com- 
mercial article  and  is  constructed  to  read  correctly  when  held  in  a 
vertical  position  and  with  the  weight  hung  on  the  hook.  When 
the  balance  is  used  in  a  horizontal  position,  the  true  tension  is 
greater  than  the  indicated  tension.  The  correction  to  be  applied 
to  the  scale  readings  is  found  by  suspending  known  weights  on  a 
cord  passing  over  a  pulley  and  secured  to  the  hook  of  the  balance 
when  held  in  a  horizontal  position.  The  thermometers  used  with 
this  apparatus  are  graduated  to  half  degrees  and  are  provided 


DESCRIPTION  OF  APPARATUS  35 


FIG.  21.    Tension  Apparatus. 


36  BASE  LINES 

with  spring  clamps  so  that  they  may  be  readily  fastened  to  the 
tape  for  making  readings,  or  removed  from  it  when  it  is  being 
carried  forward. 

24.  Marking  the  Terminal  Points. 

The  ends  of  the  base  line  to  be  measured  are  marked  in  the 
same  manner  as  triangulation  points,  that  is,  by  bolts  set  in  drill 
holes  in  stone  monuments  or  by  special  castings  set  in  concrete; 
the  points  are  tied  in  by  several  measurements  to  prevent  the 
position  being  lost.  There  is  usually  also  a  sub-surface  mark 
(see  Art.  13).  Intermediate  points  on  the  line  are  often  marked 
by  stone  or  Qoncrete  posts. 

25.  Preparation  for  the  Measurement. 

The  first  step  in  measuring  the  base  is  to  run  the  line  out 
roughly  with  transit  and  tape  and  clear  the  ground  from  obstruc- 
tions; at  the  same  time  the  measuring  stakes  are  set  in  position. 
These  may  be  4"  X  4"  stakes  set  exactly  one  tape-length  apart 
and  high  enough  so  that  the  tape  is  everywhere  clear  of  the 
ground.  On  top  of  each  stake  is  placed  a  strip  of  copper  or  zinc 
upon  which  is  scratched  the  reference  marks  used  in  making  the 
measurements.  Next,  the  slope  of  each  tape-length  is  deter- 
mined by  taking  level  readings  on  the  tops  of  all  the  stakes.  The 
intermediate  stakes  (one  or  three  in  number)  are  set  in  line,  and 
nails  for  supporting  the  tape  are  placed  at  the  proper  grade. 

26.  Measuring  the  Base.  t 
The  actual  measurement  is  begun  by  stretching  the  tape  over 

the  first  pair  of  stakes;  the  zero  end  of  the  tape  is  placed  over  the 
end  mark  of  the  base,  either  by  means  of  a  transit  set  at  one  side 
of  the  line  or  by  a  special  device  called  a  cut-ojff  cylinder.  The 
tape  is  aligned  by  means  of  field  glasses  or  by  a  transit  set  on  line 
and  the  tension  is  then  applied.  When  the  zero  graduation  of  the 
tape  is  exactly  over  the  end  mark  and  the  tension  is  correct,  the 
position  of  the  forward  (50  meter)  end  is  marked  on  the  metal 
strip,  and  the  temperature  is  read  on  all  the  thermometers.  The 
tape  is  then  carried  forward  and  the  process  repeated  until  the 
measurement  of  the  section  is  completed.  If  there  is  a  short 


CORRECTIONS  TO  BASE-LINE  MEASUREMENTS 


37 


measurement  at  the  end  of,  the  line,  this  may  be  taken  with  an 
ordinary  metric  steel  tape  graduated  its  whole  length.  When- 
ever it  is  necessary  to  set  forward  or  backward  on  one  of  the 
metal  strips  in  order  to  bring  the  reference  mark  on  the  milli- 
meter scale,  this  fact  is  recorded;  it  is  also  indicated  on  the  metal 
strips,  which  are  all  preserved  as  a  part  of  the  permanent  record. 
Measurements  of  bases  made  in  this  manner  can  be  made  at  the 
rate  of  about  2  kilometers  per  hour.  If  the  wind  is  blowing,  it 
may  be  found  necessary  to  use  three  intermediate  supports  in 
order  to  maintain  the  required  standard  of  accuracy.  If  the 
first  two  measurements  of  any  one  section  of  the  base  show  a 
discrepancy  not  exceeding  2Omm  X  VK,  the  mean  is  considered 
as  sufficiently  accurate  and  no  further  measurements  of  this 
section  are  made. 

27.  Corrections  to  Base-Line  Measurements.  —  Correction 
for  Grade. 

Where  the  slope  is  determined  by  direct  leveling,  the  most 
convenient  formula  for  computing  the  horizontal  distance  is  one 


B 


d 
FIG.  22. 


involving  the  difference  in  elevation  of  the  ends  of  the  tape. 
In  Fig.  22,  let  h  be  the  difference  in  elevation  of  the  end  points 
A  and  B,  and  let  /  be  the  length  and  d  the  required  horizontal 
distance.  Then 


Corr.  for  grade  =  Cg  =  I  —  d  =  I  — 


- 


38  BASE  LINES 


But  l-  „!-_--- 


-  *  -  /(i  -  £  -  £  .  .  .  ) 


Therefore  C. 

If  ,   h1 

-7i+8?+r--;  [I] 

If  the  slope  has  been  found  in  terms  of  the  vertical  angle  a,  the 
correction  may  be  computed  by  the  expression 

Cg  =  2  I  sin2  J  a  =  I  vers  a.  [2] 

In  good  base-line  work  the  errors  in  length  due  to  errors  in  deter- 
mining the  grade  should  never  exceed  one  part  in  one  million. 

28.  Corrections  for  Alignment. 

The  errors  in  aligning  a  straight  base-line  can  easily  be  kept  so 
small  as  to  be  negligible.  If  any  point  is  found,  however,  to  be 
out  of  line  by  an  amount  sufficient  to  affect  the  length,  the  cor- 
rection may  be  computed  by  Formula  [i], 

29.  Broken  Base. 

Sometimes  it  is  desirable  or  necessary  to  break  a  base  into  two 
parts  which  make  a  small  (deflection)  angle  with  each  other.  If 
the  two  sections  are  measured  with  the  usual  precision,  and  if  the 
angle  also  is  accurately  measured,  the  length  may  be  computed 


c 
FIG.  23. 


as  follows:  let  a  and  b,  -Fig.  23,  be  the  measured  lengths,  and  6  the 
angle  between  them,  and  let  c  be  the  desired  base,  then  from  the 
triangle  we  have 

c2  =  a2  +  b2  +  2  ab  cos  0. 

02 

Putting  for  cos  6  the  series  i h  •  •  •  ,  there  results 

2 

c2  =  (a  +  b)2  -  abd2. 


CORRECTION  FOR  ABSOLUTE  LENGTH  39 

Placing  the  factor  (a  +  b)2  outside  the  brackets  and  extracting 
the  square  root, 

abe2    "1* 
'*«*»'* -£**] 


or  c  =  a  +  b ~  —rr  (sin  i')2, 

2  (a  +  o) 

where  0  is  in  minutes  of  arc.     Substituting  the  value  of  sin  i', 

,  0602 
c  =  a  -f-  6  —  0.000,000,042,308  — — -  •  [3] 

(log.  0.000,000,042,308  =  2.62642  —  10). 

30.  Correction  for  Temperature. 

The  temperature  correction  may  be  computed  if  we  know  the 
coefficient  of  expansion,  the  actual  temperature  of  the  tape  and 
the  standard  temperature,  and  the  measured  length  of  line.  If 
k  is  the  coefficient,  t  the  observed  temperature,  /o  the  standard 
temperature,  and  L  the  measured  length,  then 

Temperature  correction  =  +kL  (t  —  fo).  [4] 

The  temperature  correction  is  often  expressed  as  a  term  in  the 
tape  equation,  as  shown  in  the  following  article. 

31.  Correction  for  Absolute  Length. 

The  length  of  the  tape  is  usually  expressed  in  the  form  of  an 
equation,  such  as 

^5i6  =  5om  +  (12.382""*  ±  0.016""*) 

-f  (0.0178""*  it  0.0007""*)  (*  -  25°-8  C.),  [5] 

meaning  that  tape  number  516  is  12.382""*  more  than  50"*  long 
at  a  temperature  of  25°. 8  C.,  and  that  0.016""*  is  the  uncertainty 
of  this  determination.  The  quantity  0.0178  is  the  temperature 
change  for  i°  for  a  50**  length,  and  0.0007  is  the  uncertainty  in 
this  number.  (The  temperature  coefficient  for  this  tape  is 
0.000,000,356.) 


BASE  LINES 


According  to  the  present  practice,  tapes  are  standardized  at 
Washington  *  under  exactly  the  same  conditions,  in  regard  to 
tension,  temperature  determination,  and  manner  of  support,  as 
those  which  are  to  govern  the  field  measurements.  By  this 
means  all  uncertainty  in  the  absolute  length  and  in  the  tension 
correction  is  kept  within  narrow  limits. 
32.  Reduction  of  Base  to  Sea-Level. 

In  order  that  all  triangulation  lines  may  be  referred  to  the 
same  surface  it  is  customary  to  employ  the  length  of  the  line  at 
sea-level  between  the  verticals  through  the  stations. 

In  Fig.  24,  let  B  represent  the  measured  base  at  elevation  h 

above  sea-level  (supposed  spherical) , 
and  b  the  length  of  base  reduced  to 
sea-level,  Ra  being  the  radius  of 
curvature  of  the  surface  (see  Art. 
97  and  Table  XI).  Then,  since 
the  arcs  are  proportional  to  their 
radii, 
b_=  Ra 

~D  ~D         I      /• 

JJ  J\a  -J-  fl 

and 


Therefore  the  reduction  to  sea-level  is 


[6] 


If  there  is  a  great  difference  in  elevation  in  different  parts  of  the 
base,  the  line  should  be  divided  into  sections  and  the  mean  value 
of  h  found  for  each  section.  Then  B  in  the  formula  is  taken  as 

*  United  States  Bureau  of  Standards,  Washington,  D.  C. 
t  See  footnote  on  page  51. 


CORRECTION  FOR  SAG  41 

the  length  of  the  section  in  question.  The  logarithm  of  the  mean 
radius  of  curvature  in  latitude  45°,  which  may  be  used  for  short 
sections,  is  6.80470. 

Question.  Is  it  necessary  to  reduce  each  triangulation  line  separately  to  sea- 
level? 

33.  Correction  for  Sag. 

Between  any  two  consecutive  points  of  support  the  tape  hangs 
in  a  curve  known  as  the  catenary,  its  form  depending  upon  the 
weight  of  the  tape,  the  tension  applied,  and  the  distance  between 
the  points  of  support. 

In  Fig.  25  let  /  be  the  horizontal  distance  between  the  supports, 
the  two  being  supposed  at  the  same  level;  let  n  be  the  number  of 


FIG.  25. 


such  spans  in  the  tape-length,  t  the  tension,  and  w  the  weight 
of  a  piece  of  tape  of  unit  length.  Also  let  v  equal  the  (vertical) 
sag  of  the  middle  point  of  the  tape  below  the  points  of  support. 
Since  the  curve  is  really  quite  flat  under  the  tension  actually 
employed  in  field-work,  the  length  of  the  catenary  will  be  sen- 
sibly equal  to  that  of  a  parabola  whose  axis  is  vertical  and  which 

passes  through  the  points  A,  B,  and  C.    The  equation  of  this 

p 
parabola  is  x2  =  —  •  y,  and  the  length  of  curve,  found  by  the 

4V 


usual  method  of  the  calculus,  is  2  s  =  I  -\ +  •  •  •  .    The 

difference  2  s  —  I  between  the  length  of  curve  AB  and  the  chord 
AB  is  approximately 

/     8     *  r  i 

25  —  /  =  -  X  —•  W 


42  BASE  LINES 

If  we  consider  the  forces  acting  on  the  tape  at  the  point  C,  and 
take  moments  about  the  point  of  support  A,  we  have 

wl      I 

—  X  -  =  v  •  t. 

2       4 

Therefore  v  =  ~  [b] 

o  t 

Substituting  in  [a]  the  value  of  v  found  in  [b],  we  find  that  the 
shortening  of  this  section  of  tape  due  to  sag  is 

8  /wP\*       I     wl\2 


For  n  sections,  we  have  nl  =  L,  whence 

Correction  for  sag  =  C8  =  —  (  —  )  •  [7] 

24  \  *  / 

34.  Tension. 

The  modulus  of  elasticity  of  the  tape  due  to  the  tension  applied 
equals  the  stress  divided  by  the  strain.  If  a  =  the  elongation 
and  L  the  length,  and  if  /  equals  the  tension  and  S  the  area  of  the 
cross  section,  then  the  modulus  of  elasticity  E  is  given  by 


LL 

Sa 
The  elongation  is 

«-C,-||,  .  [8] 

where  Ct  is  the  correction  for  the  increase  in  length  due  to  tension. 
Evidently  the  difference  in  length  due  to  a  change  from  tension 

to  to  tension  t  is  a  =  —(t  —  t0). 
oA 

The  value  of  E  must  be  found  by  trial,  applying  known  ten- 
sions and  observing  a  directly. 


TENSION  43 

To  allow  for  slight  variations  in  tension,  such  as  those  due  to 
the  failure  of  the  spring  balance  to  give  the  desired  reading  the 
instant  the  scale  of  the  tape  is  read,  the  correction  may  be  derived 
as  follows: 

Since  the  effective  length  of  the  tape  depends  both  upon  the 
elongation  due  to  tension  and  upon  the  shortening  due  to  sag, 
and  since  these  both  involve  /,  the  variation  may  be  found  by 
differentiating  the  expression 

Li  =  L  +  Ct  -  C. 


Lil_L  /^V2 

SE      24  U  /  ' 


regarding  t  as  the  independent  variable.     The  differentiation 
gives 


This  is  the  correction  due  to  small  variations  in  /.  This  quantity 
may  be  found  satisfactorily  by  actual  tests,  varying  /  by  known 
amounts  and  observing  the  change  in  length  directly. 

It  was  once  the  practice  to  compare  the  tape  with  the  standard 
when  it  was  supported  its  entire  length,  and  to  calculate  the  sag 
and  tension  corrections  to  obtain  the  effective  length  when  sup- 
ported at  a  few  points.  The  present  practice  of  comparing  the 
tape  under  the  same  conditions  that  are  to  exist  in  the  field- 
work  eliminates  all  uncertainty  in  these  computed  corrections. 

PROBLEMS 

Problem  i.  Derive  the  equation  of  the  parabola  stated  in  Art.  33.  Compute 
the  length  of  the  parabola  between  the  points  of  support  A  and  B. 

Problem  2.  The  difference  in  elevation  of  the  ends  of  a  5o-meter  tape  is  7.22  ft., 
obtained  by  leveling.  What  is  the  horizontal  distance? 

Problem  3.  A  base  line  is  broken  into  two  sections  which  meet  at  an  angle  of 
i°  59'  3i".6.  The  lengths  of  the  two  segments  are  1854.275  meters  and  3940.740 
meters.  What  is  the  distance  between  the  terminal  points? 

Problem  4.  The  length  of  a  base  line  is  i7486m.58oo  measured  at  an  altitude  of 
34.16  meters.  The  latitude  of  the  middle  point  of  the  base  is  38°  36'.  The  azi- 
muth of  the  base  is  16°  54'.  What  is  the  corresponding  length  of  the  base  at  sea- 
level? 


CHAPTER  III 

FIELD-WORK  OF  TRIANGULATION  —  MEASUREMENT 
OF  HORIZONTAL  ANGLES 

35.  Instruments  Used  in  Measuring  Horizontal  Angles. 

Instruments  intended  for  triangulation  work  are  of  two  kinds: 
the  direction  instrument,  first  designed  in  England  by  Ramsden 
in  1787,  and  the  repeating  instrument,  first  used  in  France 
about  1790.  The  former  is  the  one  chiefly  used  at  the  present 
time  for  primary  triangulation;  the  repeating  instrument,  on 
account  of  its  comparative  lightness  and  simplicity,  is  much 
used  on  triangulation  of  lesser  importance. 

Triangulation  instruments  are  larger  than  ordinary  surveying 
transits,  the  diameter  of  the  circles  varying  in  different  instru- 
ments from  8  to  30  inches.  It  is  found,  however,  that  small 
circles  can  be  graduated  so  accurately  that  little  or  nothing  is 
gained  by  using  circles  more  than  from  10  to  1 2  inches  in  diameter. 
Furthermore,  the  smaller  circles  are  less  affected  by  flexure  than 
the  larger  circles.  All  triangulation  instruments  except  the  very 
smallest  are  built  with  three  leveling  screws  and  are  used  on 
solid  supports,  like  stone  piers,  or  on  the  tripods  of  observing 
towers.  Small  instruments  intended  for  work  of  a  lower  grade 
of  accuracy  may  be  used  on  their  own  tripods. 

36.  The  Repeating  Instrument. 

The  repeating  instrument  has  an  upper  and  a  lower  plate 
arranged  exactly  as  in  the  surveyor's  transit,  and  the  graduated 
circle  is  read  by  two  or  more  verniers  graduated  to  10"  or  to  5". 
Verniers  reading  finer  than  5"  are  not  practicable,  and  depend- 
ence must  be  placed  upon  the  repetition  principle  for  securing 
greater  precision.  Fig.  26  shows  a  repeating  instrument  having 
an  8-inch  circle  which  is  read  by  two  verniers  to  10  seconds.  The 


THE  REPEATING  INSTRUMENT 


45 


FIG.  26.    Repeating  Instrument. 
(C.  L.  Berger  &  Sons.) 


46  FIELD-WORK  OF  TRIANGULATION 

telescope  of  this  instrument  has  an  aperture  of  if  inches  and  a 
magnifying  power  of  30.  Since  an  instrument  of  this  kind  is 
likely  to  be  used  in  sighting  on  pole  signals,  the  cross-hairs  are 
usually  arranged  in  the  form  of  an  X,  the  pole  bisecting  the 
angle  between  the  hairs  when  the  pointing  is  made.  Single 
vertical  hairs  would  not  be  practicable  except  on  short  lines  and 
wide  signals,  as  the  width  of  the  ordinary  hair  is  so  great  that  it 
completely  obscures  the  pole  on  long  distances. 

37.  The  Direction  Instrument. 

The  direction  instrument  has  but  one  horizontal  circle,  read 
by  two  or  more  microscopes  instead  of  verniers.  The  circle  can 
be  turned  about  the  axis  and  clamped  in  any  desired  position. 
The  motion  of  the  telescope  and  the  microscopes  is  entirely  in- 
dependent of  the  motion  of  the  circle;  the  latter  can  be  shifted 
while  the  upper  part  of  the  instrument  (called  the  alidade)  re- 
mains clamped.  It  is  evident  that  a  repeater  could  be  used  as  a 
direction  instrument,  but  that  a  direction  instrument  could  not 
be  used  for  measuring  angles  by  the  repetition  method-  Fig.  27 
shows  a  i2-inch  theodolite  with  microscopes  reading  to  seconds. 

The  circle  of  the  direction  instrument  is  usually  graduated  into 
5'  spaces.  The  direction  of  the  line  of  sight  of  the  telescope  is 
read  by  first  noting  the  degrees  and  5'  spaces  in  a  small  index 
microscope,  and  then  accurately  measuring  the  fractional  parts 
of  the  5'  spaces  by  means  of  the  three  equidistant  micrometer 
microscopes.  The  micrometers  can  usually  be  read  to  seconds 
directly,  and  to  tenths  of  a  second  by  estimation.  The  mean 
of  the  three  micrometer  readings  is  taken  as  the  true  reading, 
and  this  is  added  to  the  reading  of  the  index  microscope  to 
obtain  the  direction. 

The  telescope  of  the  1 2-inch  theodolite  used  by  the  Coast  Sur- 
vey has  an  aperture  of  2.4  inches,  a  focal  length  of  29  inches,  and 
magnifying  powers  of  30,  45,  and  60.  The  circle  is  graduated 
to  5'  and  reads  to  seconds  by  means  of  three  microscopes. 
A  camel's-hair  brush  (inside  the  cover  plate)  sweeps  over  the 
graduations.  The  base  is  made  very  heavy,  and  the  bearing 


THE  DIRECTION  INSTRUMENT 


47 


FIG.  27.     Twelve-inch  Theodolite. 
(Coast  and  Geodetic  Survey.) 


48 


FIELD-WORK   OF   TRIANGULATION 


surfaces  of  the  centers  are  glass  hard.  The  centers  on  this  instru- 
ment are  very  long.  The  upper  parts  of  the  instrument  are 
made  chiefly  of  aluminum,  in  order  to  diminish  the  weight 
bearing  upon  the  centers.  This  design  produces  an  instrument 
of  exceptional  stability. 

Direction  instruments  are  used  chiefly  on  long  lines  and  in 
connection  with  heliotropes  or  lights.  For  this  reason  the  cross- 
hairs usually  consist  of  two  vertical  hairs,  set  so  as  to  subtend 
an  angle  of  from  10"  to  20",  and  two  horizontal  hairs,  set  much 
farther  apart  and  used  merely  to  limit  the  portion  of  the  vertical 
hairs  to  be  used  in  pointing. 

38.   The  Micrometer  Microscope. 

The  construction  of  the  micrometer  microscope  is  shown  in 
Fig.  28.  The  head  of  the  screw  is  graduated  into  60  divisions 


E-S 


FIG.  28. 

corresponding  to  seconds  of  angle.  As  the  screw  head  is  turned 
the  two  parallel  hairs  in  the  field  of  the  microscope  are  moved  in 
a  direction  parallel  (tangent)  to  the  edge  of  the  graduated  circle. 
The  distance  between  these  hairs  is  just  sufficient  to  leave  a 
small  white  space  on  each  side  of  a  line  of  graduation  when  it  is 
centered  between  the  two  hairs.  The  pitch  of  the  screw  and  the 
focal  length  of  the  objective  of  the  microscope  are  so  related  that 
five  whole  turns  of  the  screw  will  carry  the  hairs  from  one  gradua- 
tion to  the  next.  The  number  of  whole  turns  of  the  screw  may 


RUN  OF  THE  MICROMETER  49 

be  counted  on  a  notched  scale  visible  in  the  field  of  view  of  the 
microscope.  The  fraction  of  a  space  to  be  measured  is  that  lying 
between  the  zero  point  of  the  notched  (comb)  scale  and  the 
graduated  line  last  passed  over  by  the  zero  point.  Strictly 
speaking,  the  zero  point  is  that  position  of  the  hairs  in  the  zero 
notch  at  which  the  scale  on  the  screw  head  will  read  exactly  zero. 
The  position  of  the  hairs  for  a  zero  reading  of  the  screw  may  be 
adjusted  by  holding  fast  the  graduated  ring  on  the  screw  head 
and  turning  the  milled  edge  screw  head  which  moves  the  hairs. 
The  microscope  inverts  the  image  of  the  graduated  circle  so  that 
graduations  increasing  in  the  direction  of  azimuths  will  appear 
to  increase  from  left  to  right  in  the  field  of  view  of  the  microscope. 
The  readings  on  the  screw  head  increase  as  the  screw  is  turned 
left-handed,  and  the  hair  lines  move  in  the  direction  of  decreasing 
graduations  over  the  circle. 

To  measure  the  space  between  the  zero  of  the  microscope  and 
the  last  line  passed  over,  it  is  only  necessary  to  turn  the  screw 
until  the  graduation  in  question  bisects  the  space  between  the 
hairs,  and  then  to  read  the  comb  scale  and  the  scale  on  the  screw 
head.  This  reading  is  to  be  added  to  the  number  of  the  gradu- 
ated line,  to  obtain  the  direction  as  shown  by  this  microscope. 
For  example,  if  the  screw  is  turned  two  revolutions  (two  notches) 
and  ten  divisions  in  order  to  center  the  47°  05'  mark  between 
the  hairs,  the  reading  of  this  microscope  is  47°  05'  +  2'  10"  = 
47°  of  10".  A  complete  set  of  readings  of  one  direction  would 
consist  of  readings  of  each  of  the  three  microscopes  on  both  the 
preceding  and  the  following  graduations,  six  readings  in  all. 

39.  Run  of  the  Micrometer. 

If  the  microscope  is  perfectly  adjusted  with  respect  to  the 
graduated  circle,  and  if  the  latter  is  perfectly  plane,  then  five 
whole  revolutions  of  the  screw  should  carry  the  hairs  from  one 
line  to  the  next,  and  the  reading  of  the  screw  should  be  the  same 
on  all  lines.  Since  this  condition  is  rarely  fulfilled,  there  is  ordi- 
narily a  small  difference  in  the  forward  and  backward  readings, 
called  the  error  of  run  of  the  micrometer. 


50  FIELD-WORK  OF  TRIANGULATION 

The  forward  reading  F  is  the  reading  taken  when  the  threads 
are  moved  from  the  zero  position  (Fig.  29)  to  the  preceding  mark 
(25'  in  Fig.  29a).  The  back  reading  B  is  the  one  taken  on  the 
following  (30')  mark,  Fig.  29b.  The  graduations  on  the  screw- 
head  decrease  as  the  threads  move  from  25'  to  30'.  If  the 


FIG.  29.     Field  of  Micrometer  Microscope. 


FIG.  2ga.     Forward  Reading. 


FIG.  290.     Back  Reading. 


micrometer  screw  is  turned  so  that  the  threads  move  from  its 
zero  to  the  25'  mark,  then  the  reading  F  is  to  be  added  directly 
to  the  circle  reading.  In  the  figure  the  reading  is  201°  25'  + 
i'  26.2"  =  201°  26'  26.2".  Without  assuming  anything  in  regard 
to  the  actual  value  of  one  turn  of  the  screw,  the  value  may  be 


RUN  OF  THE  MICROMETER  51 

computed  by  dividing  the  angular  space  between  graduations  by 
the  number  of  turns  or  divisions  recorded  in  passing  from  one 
graduation  to  the  next.  If  R  =  the  value  of  one  revolution,  then 

R_         300"         =    300" 
300  +  F  —  B      300  +  r 

where  r  is  the  run  of  the  micrometer  in  seconds  (divisions)  as 
indicated  by  the  differences  of  the  forward  and  backward  read- 
ings, positive  if  F  is  greater  than  B.  If  the  screw  turns  more 
than  five  times  in  passing  from  the  25'  line  to  the  30'  line,  the 
reading  B,  on  the  30'  line,  will  be  smaller  than  F,  since  the  screw 
readings  are  decreasing.  This  makes  F  —  B  =  r  positive. 
Hence  the  denominator  of  the  above  fraction  is  greater  than  300, 
and  the  value  of  one  turn  is  less  than  unity,  as  it  should  be  ac- 
cording to  the  assumption. 

If  F  is  the  forward  reading  in  any  given  case,  it  must  be  con- 
verted into  arc  by  multiplying  it  by  the  value  of  one  turn,  since 
F  is  simply  a  certain  number  of  turns  and  divisions,  not  the  true 
number  of  minutes  and  seconds, 

Therefore  True  reading  =  F  (- 

\3oo 

Since  r  is  small  (say  2"  or  3"),  it  is  permissible  to  write 
R  = 


300 

and  the  true  reading  =  F  ( i  —  -.  .  .  j 

=  F-F  —  -  (a) 

300 

This  formula  corrects  the  forward  reading  only,  and  assumes 

*  By  actual  division  =i  —  x  +  xz  —  x3  +  -  •  •  .     If*  is  small  enough 

so  that  x*  and  the  following  term  may  be  neglected,  then  -     —  =  i  —  x. 


52  FIELD-WORK   OF  TRIANGULATION 

that  the  bisection  and  reading  are  perfectly  made.  If  the  back 
reading  is  corrected  in  a  similar  manner,  the  result  is 

True  reading  =  300"  -  (300"  -  B)  (i  -  - 

The  first  300"  is  the  space  between  25'  and  30'.  The  factor 
(300"  —  B)  is  the  space  between  zero  and  the  30'  mark  deter- 
mined by  the  B  reading.  It  should  be  remembered  that  when 
the  micrometer  is  turned  to  the  30'  mark,  the  readings  are  de- 
creasing; therefore  the  direct  reading  does  not  give  this  space, 
but  5  minus  this  space.  Simplifying  this  expression  we  have 

True  reading  =  B  +  r  -  B  —•  (b) 

300 

Since  there  is  no  reason  for  preferring  either  the  forward  or  the 
backward  reading,  the  mean  is  used  as  the  best  value.  The 
mean  of  (a)  and  (b)  is 

F  +  B      r      F  +  B     '   r 
-  -r  -  -  x  — 

2  2  2  300 

•p    IT? 
If  -   — —  =  m,  then  the  correction  to  m,  the  mean  of  the  two 

2 

readings,  is 

Corr.  = m [10] 

2          300 

A  general  table  may  be  computed  for  different  values  of  m  and 
r,  so  that  no  special  computation  is  necessary  when  correcting  a 
direction.  It  is  good  practice  to  determine  r  from  all  the  F  and 
B  readings,  and  to  employ  this  average  value  when  making  the 
corrections. 

40.  Vertical  Collimator. 

In  centering  a  signal  over  a  station,  placing  a  mark  under  a  new 
signal,  or  centering  the  theodolite  over  the  station  mark,  the 
Coast  Survey  observers  sometimes  employ  the  vertical  collimator 
shown  in  Fig.  30.  The  instrument  is  adjusted  by  means  of  spirit 
levels,  which  revolve  around  a  vertical  axis  Like  those  of  a  transit. 


ADJUSTMENTS   OF   THE   THEODOLITE 


53 


A  telescope  may  be  placed  in  coincidence  with  the  vertical  axis 
of  the  collimator,  and  its  line  of  sight  adjusted  to  point  vertically 
downward.  With  the  instrument  in  this  position  the  observer 
may  obtain  a  point  which  is  vertically  above  the  center  mark  of 
the  station. 


FIG.  30.    Vertical  CoIIimator. 
(Coast  and  Geodetic  Survey.) 


41.  Adjustments  of  the  Theodolite. 

The  adjustment  of  the  levels  attached  to  the  alidade  is  made 
by  means  of  reversals  about  the  vertical  axis  of  the  instrument, 
exactly  as  with  the  engineer's  transit. 

The  adjustment  of  the  stride  level  is  tested  by  placing  it  on  the 
horizontal  axis,  reading  both  ends  of  the  bubble,  and  then  re- 
versing the  level  and  reading  again.  The  adjusting  screws  of 
the  stride  level  should  be  turned  so  that  the  bubble  moves  half- 


54  FIELD-WORK  OF  TRIANGULATION 

way  back  from  the  second  position  to  the  first.  When  the 
stride  level  is  so  adjusted  that  it  reads  the  same  in  either  position, 
it  is  in  correct  adjustment,  and  the  horizontal  rotation  axis  may 
then  be  leveled  by  moving  the  adjustable  end  of  the  axis  until 
the  bubble  is  in  the  center  of  its  tube.  Of  course  the  two  adjust- 
ments may  be  made  simultaneously.  If  desired,  the  stride  level 
may  be  used  also  to  make  the  vertical  axis  truly  vertical. 

The  adjustment  of  the  line  of  sight  in  a  plane  perpendicular 
to  the  horizontal  axis  may  be  made  by  reversals  about  the  hori- 
zontal axis  as  in  testing  an  engineer's  transit;  or  it  may  be  made 
by  sighting  an  object,  lifting  the  telescope  out  of  its  bearings,  and, 
after  reversing  the  axis,  replacing  it  in  the  bearings.  If  the 
object  is  no  longer  in  the  line  of  sight,  the  reticle  is  brought  half- 
way back  from  the  second  position  toward  the  first. 

The  test  of  the  adjustment  of  the  microscopes  is  made  by 
measuring  the  run  of  each  micrometer,  taking  first  a  forward  and 
then  a  back  reading.     In  case  the  run  of  a  microm- 
eter is  greater  than  about  3",  it  should  be  adjusted 
by  changing  the  distance  from  the  objective  to  the 
reticle,  and  then  moving  the  whole  microscope  so 
that  the  graduations  are  again  in  focus.     If  the 
image  of  the  division  is  greater  than  5  whole  turns 
of  the  screw,  the  objective  should  be  moved  toward 
the  eyepiece,  and  then  the  whole  microscope  moved 
away  from  the  circle.     Moving  the  objective  away 
FIG.  31.       from  the  micrometer  lines  diminishes  the  angle  be- 
tween the  two  lines  of  sight  corresponding  to  the  5 
turns,  and  reduces  the  size  of  the  image  of  the  division  (Fig.  31). 
It  will  usually  require  a  series  of  trials  to  perfect  this  adjust- 
ment. 

42.  Effect  of  Errors  of  Adjustment  on  Horizontal  Angles. 

The  effect  of  errors  due  to  the  inclination  of  the  horizontal  axis 

to  the  horizon,  and  those  due  to  the  imperfect  adjustment  for 

collimation  (line  of  sight),  are  not  independent  of  each  other. 

These  errors  are  usually  so  small,  however,  that  it  is  permissible 


EFFECT  OF  ERRORS  OF  ADJUSTMENT 


55 


to  compute  their  effect  separately,  as  though  only  one  existed  at 
one  tune.  In  Fig.  32,  Z  is  the  true  zenith  and  Z'  the  point  where 
the  vertical  axis  of  the  instrument  prolonged  pierces  the  celestial 
sphere.  5  is  a  point  whose  altitude  is  h.  Assuming  that  the 
horizontal  axis  makes  an  angle  i  with  the  horizon,  and  that  all 
other  errors  are  zero,  then  from  the  figure  it  will  be  seen  that  we 

may  write 

sinZ'      si 


n 


_ 

sin  i       sin  Z'S  ' 
or,  with  sufficient  accuracy, 

where  h  is  the  angular  altitude  of  the  point  sighted. 


FIG.  32. 


It  appears,  then,  that  for  each  point  sighted  there  should  be  a 
correction  to  the  circle  reading  equal  to  i  tan  h.  Triangulation 
points  are  usually  so  nearly  on  the  horizon,  and  by  careful  atten- 
tion to  the  leveling  the  error  i  may  easily  be  kept  so  small,  that 
there  is  seldom  any  necessity  for  applying  the  correction  except 
for  such  observations  as  those  on  a  circumpolar  star  for  azimuth. 

In  the  preceding  paragraph  it  is  assumed  that  the  vertical  axis 
is  truly  vertical,  the  graduated  circle  being  horizontal,  while  the 
horizontal  axis  is  not  horizontal.  If  the  two  axes  are  at  right 
angles  to  each  other,  but  the  vertical  axis  is  inclined  to  the  true 
vertical  by  a  small  angle  i,  it  may  be  shown,  by  a  diagram  similar 
to  Fig.  32,  that  the  same  correction  applies  to  this  case  also. 


56  FIELD-WORK   OF  TRIANGULATION 

The  error  of  a  horizontal  direction  due  to  an  error  of  collima- 
tion  may  be  computed  as  follows :  Let  the  error  in  the  sight  line 
be  represented  by  c\  then,  when  the  axis  of  collimation  (Fig.  33) 
traces  out  the  great  circle  ZN,  the  line  of  sight  traces  out  the 
parallel  circle  SA ,  which  is  c  seconds  from  ZN.  If  S  be  any  point 


FIG.  33. 

toward  which  the  cross-hair  is  pointing,  and  if  arc  SN  be  drawn 
perpendicular  to  ZN,  the  error  in  direction,  or  the  angle  at  Z,  is 
found  from  the  equation 

sinZ  _     sine 

sin  N      sin  ZS ' 
or,  since  Z.N  =  90°,  Z  =  c  sec  h.  [12] 

Each  direction  should  therefore  be  corrected  by  the  quantity 
c  sec  h.  On  account  of  the  small  value  of  c  in  a  well-adjusted 
instrument  this  correction  is  necessarily  small;  furthermore,  it  is 
usually  eliminated  from  the  final  result  by  the  method  employed 
in  making  the  observations. 

43.   Method  of  Measuring  the  Angles. 

In  measuring  angles  with  the  repeating  instrument  the  common 
practice  has  been  to  measure  the  angle  six  times,  beginning  with 
the  left-hand  signal  of  a  pair  and  measuring  toward  the  right,  and 
then,  after  reversing  the  telescope  both  in  altitude  and  in  azimuth, 
to  measure  six  times  from  right  to  left.  The  recent  practice  of 
the  Coast  Survey  has  been  to  measure  first  the  angle  itself  by 
six  repetitions,  left  to  right,  with  the  telescope  direct,  then  the 


METHOD  OF  MEASURING  THE  ANGLES  57 

explement  (360°  minus  the  angle)  six  times,  moving  the  alidade 
in  the  same  direction  as  before,  left  to  right,  the  telescope  being 
reversed.  This  brings  the  vernier  nearly  back  to  the  same  read- 
ing as  by  the  previous  method,  but  it  differs  in  the  mechanical 
operation.  If  there  is  any  systematic  effect  on  the  angle,  due  to 
the  action  of  clamps  or  to  drag  on  the  centers,  it  is  eliminated 
from  the  final  result,  provided  such  errors  are  the  same  for  a 
large  as  for  a  small  angle. 

The  reversal  of  the  telescope  in  the  preceding  programs  is  in- 
tended to  eliminate  the  errors  of  adjustment  of  the  line  of  colli- 
mation  and  of  the  rotation  axis  of  the  telescope.  It  does  not 
eliminate  errors  due  to  imperfect  leveling.  The  measurement 
of  angles  in  both  the  left-to-right  and  the  right-to-left  direction 
is  designed  to  eliminate  possible  twist  in  the  support  of  the  instru- 
ment, upon  the  assumption  that  this  twist,  takes  place  at  a 
uniform  rate. 

In  order  to  eliminate  errors  due  to  faulty  graduation  of  the 
circle,  the  initial  reading  for  different  sets  of  observations  may 

s     O 

be  shifted  by  ^ — .  where  m  is  the  number  of  sets  taken  and  n 
mn 

is  the  number  of  verniers.  For  example,  in- taking  four  sets  with 
a  two-vernier  instrument,  the  vernier  would  be  set  ahead  45° 
each  time.  Errors  in  the  graduation  of  the  verniers  may  be 
eliminated  in  a  similar  manner  by  changing  the  vernier  setting 

—  th  part  of  a  circle  division  at  the  beginning  of  each  new  set. 
m 

For  four  sets,  on  a  10'  graduation,  the  first  setting  might  be  zero, 
the  second  45°  02'  30",  the  third  90°  05'  oo",  and  the  fourth 

135°  07' 30". 

With  the  direction  instrument  the  method  of  measurement 
consists  in  first  pointing  the  telescope  at  some  conspicuous  signal, 
selected  as  the  first  of  the  series  around  the  horizon,  and  reading 
all  the  microscopes,  then  turning  the  telescope  to  the  other  signals 
in  order  and  reading  all  the  microscopes  at  each  pointing.  After 
the  last  pointing  has  been  completed  and  the  microscopes  read, 


58  FIELD-WORK  OF  TRIANGULATION 

the  telescope  is  reversed,  the  pivots  remaining  in  the  same  bear- 
ings, and  the  series  is  repeated,  the  signals  being  sighted  in  the 
reversed  order.  The  horizontal  circle  remains  clamped  during 
the  entire  process.  The  above  measurements  constitute  a  single 
"set."  As  many  sets  may  be  taken  as  are  required  to  give  the 
necessary  accuracy.  To  eliminate  systematic  errors  of  gradua- 
tion and  errors  of  the  micrometers,  the  circle  reading  is  advanced 
for  each,  new  set,  as  explained  later  in  the  "  Instructions  for 
Primary  Triangulation."  It  should  be  observed  that  the  ac- 
curacy depends  upon  the  circle's  remaining  undisturbed  in 
azimuth  during  each  set. 

In  making  bisections,  either  when  pointing  on  the  signal  or 
when  reading  the  microscopes,  the  observer  should  proceed  as 
rapidly  as  he  can  without  making  careless  pointings  and  without 
danger  of  making  mistakes.  Much  time  spent  in  perfecting 
settings  and  in  watching  them  to  see  if  they  are  correct  appears 
to  reduce  slightly  the  accidental  errors  of  observation,  but  does 
not  really  increase  the  accuracy  of  the  work,  as  shown  by  the 
final  results  of  the  triangulation.  The  longer  the  time  that  is 
permitted  to  intervene  between  pointings,  the  greater  the  oppor- 
tunity for  the  circle  to  shift  its  position  or  change  its  temperature ; 
and  the  effects  of  these  changes  are  probably  greater  than  the 
accidental  errors  of  pointing  and  reading. 

44.   Program  for  Measuring  Angles. 

'Various  programs  of  observations  have  been  devised,  with  a 
view  to  eliminating  or  reducing  the  errors  in  horizontal  angles. 
The  principal  errors  which  have  to  be  considered  in  planning  the 
field  work  of  the  triangulation,  and  the  methods  adopted  for 
eliminating  them,  are  as  follows: 

i.  Errors  due  to  non-adjustment  of  the  theodolite.  These  are 
all  eliminated  by  the  use  of  the  instrument  in  the  direct  and 
reversed  positions,  except  that  due  to  erroneous  leveling.  The 
leveling  of  the  plate  and  the  horizontal  axis  must  be  carefully 
attended  to  in  setting  up  the  instrument,  and  must  be  corrected 
whenever  it  becomes  necessary.  This  may  be  done  at  any  time 


PROGRAM   FOR  MEASURING  ANGLES  59 

between  sets  of  angles  or,  if  a  repeater  is  used,  at  any  time  when 
the  lower  clamp  is  loose. 

2.  Errors  arising  from  imperfections  of  graduation.     These  are 
practically  eliminated  by  distributing  the  readings  uniformly 
around  the  circle. 

3.  Errors  of  eccentricity  of  the  circle  and  alidade.    These  errors 
are  almost  wholly  eliminated  by  reading  two  or  more  equidis- 
tant microscopes  or  verniers. 

4.  Errors  due  to  twisting  of  the  tripod  under  the  action  of  the  sun's 
rays.    The  twist  is  eliminated  by  reversing  the  direction  of  the 
measurements,  provided  the  rate  of  twist  and  the  speed  of 
measuring  the  angles  are  both  uniform.     The  rate  of  twist  on 
some  towers  has  been  found  to  be  about  one  second  of  angle  per 
minute  of  time.     On  the  slender  towers  used  on  the  gSth  meridian 
triangulation,  the  twist  was  so  small  that  it  could  not  be  detected 
by  an  examination  of  the  measurements. 

5.  Errors  due  to  irregular  refraction  of  the  atmosphere  and  to 
difficult  seeing.     These  will  be  partly  eliminated  by  taking  a  large 
number  of  measurements;  if  the  results  indicate  that  the  neces- 
sary precision  is  not  being  obtained,  it  will  be  best  to  wait  until 

favorable.    This  can  be  judged  best  by 


the  "  probable  error  "  of  the  direction. 

6.  The  personal  error  of  the  observer.    The  personal  error  is 
partly  eliminated  by  measuring  the  angle  a  large  number  of  tunes. 

7.  Errors  due  to  temperature  and  wind.     Errors  due  to  fluctua- 
tion of  the  temperature  of  the  instrument,  and  to  vibrations 
caused  by  wind,  may  be  reduced  by  shielding  the  instrument 
from  the  sun  and  wind,  either  by  a  tent  or  by  a  temporary  build- 
ing. 

The  following  list  of  instructions  to  observers  is  taken  from 
the  Coast  and  Geodetic  Survey  Special  Publication  No.  19 
(1914),  and  represents  the  present  practice  of  that  Survey. 

i.  Instruments.  —  In  general,  direction  instruments  of  the  highest  grade  should 
be  used  in  triangulation  of  this  class.  Repeating  theodolites  are  to  be  used  only 
when  the  station  to  be  occupied  is  in  such  a  position  as  to  be  difficult  of  occupation 


6o 


FIELD-WORK   OF  TRIANGULATION 


with  a  direction  instrument  or  when  there  is  doubt  of  the  instrument  support  being 
of  such  a  character  as  to  insure  that  the  movement  of  the  observer  about  the  in- 
strument does  not  disturb  it  in  azimuth.  Such  stations  usually  occur  on  lighthouses 
and  buildings. 

2.  Number  of  observations  —  Main  scheme  —  Direction  instrument.  —  In  making 
the  measurements  of  horizontal  directions  measure  each  direction  in  the  primary 
scheme  16  times,  a  direct  and  reverse  reading  being  considered  one  measurement, 
and  1 6  positions  of  the  circle  are  to  be  used,  corresponding  approximately  to  the 
following  readings  upon  the  initial  signal: 


Number. 

Reading. 

Number. 

Reading. 

I 

O  OO  40 

9 

128  oo  40 

2 

3 

IS  01  SO 
30  03  10 

10 

ii 

143  oi  50 
158  03  10 

4 
6 

45  04  20 
64  oo  40 
79  oi  5o 

12 
13 
14 

173  04  20 

192  oo  40 
207  oi  50 

8 

94  03  10 
109  04  20 

15 

16 

222  03  10 
237  04  20 

3.  When  a  broken  series  is  observed,  the  missing  signals  are  to  be  observed  later 
in  connection  with  the  chosen  initial  or  with  some  other  one,  and  only  one,  of  the 
stations  already  observed  in  that  series.     With  this  system  of  observing  no  local 
adjustment  is  necessary.     Little  time  should  be  spent  in  waiting  for  the  doubtful 
signal  to  show.     If  it  is  not  showing  within,  say,  one  minute  of  when  wanted,  pass 
to  the  next.     A  saving  of  time  results  from  observing  many  or  all  of  the  signals  in 
each  series,  provided  there  are  no  long  waits  for  signals  to  show,  but  not  otherwise. 

4.  Standard  of  accuracy.  —  In  selecting  the  conditions  under  which  to  observe 
primary  directions,  proceed  upon  the  assumption  that  the  maximum  speed  con- 
sistent with  the  requirement  that  the  closing  error  of  a  single  triangle  in  the  primary 
scheme  shall  seldom  exceed  three  seconds,  and  that  the  average  closing  error  shall 
be  but  little  greater  than  one  second,  is  what  is  desired  rather  than  a  greater  ac- 
curacy than  that  indicated  with  slower  progress.    This  standard  of  accuracy  used 
in  connection  with  other  portions  of  these  instructions  denning  the  necessary 
strength  of  figures  and  frequency  of  bases  will  in  general  insure  that  the  probable 
error  of  any  base  line,  as  computed  from  an  adjacent  base,  is  about  i  part  in  88,000, 
and  that  the  actual  discrepancy  between  bases  is  always  less  than  i  part  in  25,000. 

5.  Rejections  —  Direction  observations.  —  The  limit  for  rejection  of  observations 
upon  directions  in  the  main  scheme  shall  be  5  seconds  from  the  mean.     No  observa- 
tion agreeing  with  the  mean  within  this  limit  is  to  be  rejected  unless  the  rejection  is 
made  at  the  time  of  taking  the  observation  and  fof  some  other  reason  than  simply 
that  the  residual  is  large.     A  new  observation  is  to  be  substituted  for  the  rejected 
one  before  leaving  the  station,  if  possible  without  much  delay. 


ii.    Vertical  measures  in  main  scheme.  —  At  each  station  in  the  main  schema 


PROGRAM   FOR  MEASURING  ANGLES  6l 

vertical  measures  are  to  be  made  over  all  lines  in  the  main  scheme  radiating  from  it. 
These  vertical  measures  should  be  made  on  as  many  days  as  possible  during  the 
occupation  of  the  station,  but  in  no  case  should  the  occupation  of  the  station  be  pro- 
longed in  order  to  secure  such  measures.  Three  measures,  each  with  the  telescope 
in  both  the  dire.ct  and  the  reversed  positions,  on  each  day,  are  all  that  are  required. 
These  measures  may  be  made  at  any  time  between  11.00  A.M.  and  4.30  P.M.,  except 
that  in  no  case  should  primary  vertical  measures  be  made  within  one  hour  of  sun- 
set. It  is  desirable,  however,  with  a  view  of  avoiding  errors  due  to  diurnal  varia- 
tion of  refraction,  to  have  a  fixed  habit  of  observing  the  verticals  in  the  main  scheme 
at  a  certain  hour,  as,  for  example,  between  2  and  3  P.M.  If  the  vertical  measures  at 
a  station  are  made  by  the  micrometric' method,  double  zenith  distance  measures 
shall  be  made  on  at  least  two  of  the  lines  radiating  from  that  station. 

13.  Marking  of  stations.  —  Every  station,  whether  it  is  in  the  main  scheme  or  is 
a  supplementary  or  intersection  station,  which  is  not  in  itself  a  permanent  mark,  as 
are  lighthouses,  church  spires,  cupolas,  towers,  large  chimneys,  sharp  peaks,  etc., 
shall  be  marked  in  a  permanent  manner.     At  least  one  reference  mark  of  a  perma- 
nent character  shall  be  established  not  less  than  10  meters  from  each  station  of  the 
main  scheme  and  accurately  referred  to  it  by  a  distance  and  direction.     Such  ref- 
erence marks  shall  preferably  be  established  on  fence  or  property  lines,  and  always 
in  a  locality  chosen  to  avoid  disturbance  by  cultivation,  erosion,  or  building.     It  is 
desirable  to  establish  such  reference  marks  at  all  marked  stations.     At  all  stations 
where  digging  is  feasible  both  underground  and  surface  marks  which  are  not  in  con- 
tact with  each  other  shall  be  established.  '  Wood  is  not  to  be  used  in  permanent 
marks.  .      • 

14.  Descriptions  of  stations.  —  Descriptions  shall  be  furnished  of  all  marked 
stations.     For  each  station  which  is  in  itself  a  mark,  as  are  lighthouses,  church 
spires,  cupolas,  towers,  large  chimneys,  sharp  peaks,  etc.,  either  a  description  must 
be  furnished,  or  the  records,  lists  of  directions,  and  lists  of  positions  must  be  made 
to  show  clearly  in  connection  with  each  point  by  special  words  or  phrases  if  neces- 
sary the  exact  point  of  the  structure  or  object  to  which  the  horizontal  and  vertical 
measures  refer.     Every  land  section  corner  connected  with  the  triangulation  must 
be  fully  described.    The  purpose  of  the  description  is  to  enable  one  who  is  un- 
familiar with  the  locality  to  find  the  exact  point  determined  as  the  station  and  to 
know  positively  that  he  has  found  it.     Nothing  should  be  put  into  the  description 
that  does  not  serve  this  purpose.     A  sketch  accompanying  the  description  should 
not  be  used  as  a  substitute  for  words.     All  essential  facts  which  can  be  stated  in 
words  should  be  so  stated,  even  though  they  are  also  shown  in  the  sketch. 

15.  Abstracts  and  duplicates.  —  The  field  abstracts  of  horizontal  directions  and 
vertical  measures  are  to  be  kept  up  and  checked  as  the  work  progresses,  and  all  notes 
as  to  eccentricities  of  signals  or  instrument,  of  height  of  point  observed  above 
ground,  etc.,  which  are  necessary  to  enable  the  computation  to  be  made,  are  to  be 
incorporated  in  the  abstracts.     As  soon  as  each  volume  of  the  original  record  has 
been  fully  abstracted  and  the  abstracts  checked,  it  is  to  be  sent  to  the  Office,  the 
corresponding  abstracts  being  retained  by  the  observer.     A  duplicate  of  the  de- 
scription of  stations  is  to  be  made.     If  the  original  descriptions  of  stations  are 


62  FIELD-WORK  OF  TRIANGULATION 

written  in  the  record  books,  a  copy  of  these  descriptions  compiled  in  a  separate 
book  may  be  considered  the  duplicate  and  should  then  be  marked  as  such.  A 
duplicate  of  the  miscellaneous  notes  mentioned  above  may  also  be  made  if  con- 
sidered desirable.  No  other  duplicates  of  the  original  records  are  to  be  made. 
Pencil  originals  should  not  be  inked  over. 

16.  Number  of  observations  —  Main  scheme  —  Repeating  theodolite.  —  If  a  te- 
peating  theodolite  is  used  for  observations  in  the  main  scheme,  corresponding  to 
those  indicated  in  paragraph  2,  make  the  observations  in  sets  of  six  repetitions  each. 
'  For  each  angle  measured  follow  each  set  of  six  repetitions  upon  an  angle  with  the 
telescope  in  the  direct  position  immediately  by  a  similar  set  of  six  on  the  explement 
of  the  angle  with  the  telescope  in  the  reversed  position.  It  is  not  necessary  to  re- 
verse the  telescope  during  any  set  of  six.  Make  the  total  number  of  sets  of  six 
repetitions  on  each  angle  ten  —  five  directly  on  the  angle  and  five  on  its  explement. 
Measure  only  the  single  angles  between  adjacent  lines  of  the  primary  scheme  and 
the  angle  necessary  to  close  the  horizon.  With  this  scheme  of  observing  no  local 
adjustment  is  necessary,  except  to  distribute  the  horizon  closure  uniformly  among 
the  angles  measured.  The  limit  of  rejection  corresponding  to  that  stated  in  para- 
graph 5  shall  be  for  a  set  of  six  repetitions  4"  from  the  mean. 

19.  Field  computations.  —  The  field  computations  are  to  be  carried  to  hun- 
dredths  of  seconds  in  the  angles,  azimuths,  latitudes,  and  longitudes,  and  to  seven 
places  in  the  logarithms.  The  field  computation  may  be  stopped  with  the  com- 
pletion of  the  lists  of  directions  for  all  stations  and  objects,  and  the  triangle  side 
computation  for  the  main  scheme  and  supplementary  stations,  unless  there  are 
special  reasons  for  carrying  it  further.  The  computation  to  this  point  should  be 
kept  up  as  closely  as  possible  as  the  work  progresses,  to  enable  the  observer  to  know 
that  the  observations  are  of  the  required  degree  of  accuracy.  No  least  square  ad- 
justments are  to  be  made  in  the  field.  All  of  the  computation,  taking  of  means, 
etc.,  which  is  done  in  the  record  books  and  the  lists  of  directions  should  be  so 
thoroughly  checked  by  some  person  other  than  the  one  who  originally  did  it  as  to 
make  it  unnecessary  to  examine  it  in  the  Office.  The  initials  of  the  person  making 
and  checking  the  computations  in  the  record  books  and  the  lists  of  directions  should 
be  signed  to  the  record  as  the  computation  and  checking  progress. 

Investigations  of  the  accumulated  error  in  the  azimuth  of  a 
chain  of  triangles  indicate  that  there  is  a  systematic  tendency 
of  the  triangulation  to  twist  in  azimuth,  due  to  unequal  heating 
of  the  different  parts  of  the  theodolite  by  the  sun.  In  day  ob- 
servations on  arcs  running  north  and  south  there  appears  to  be  a 
greater  accumulated  error  in  azimuth  on  the  east  side  of  the  chain 
than  on  the  west  side.  This  is  apparently  due  to  the  fact  that 
the  observations  were  made  chiefly  or  wholly  in  the  afternoon. 
Observations  made  at  night  show  less  difference  between  the  two 


PROGRAM   FOR  MEASURING  ANGLES 


sides  of  a  chain  of  triangles.  The  errors  due  to  this  cause  may  be 
diminished  by  making  the  instrument  out  of  metal  having  a 
lower  coefficient  of  expansion,  such  as  nickel-iron,  and  by  in- 
creasing the  proportion  of  night  observations.  The  unequal 
heating  effect  may  also  be  diminished  in  day  observations  by 
turning  the  circle  180°  in  azimuth  between  sets.  The  following 
set  of  pointings,  to  be  substituted  for  that  on  p.  60,  is  designed 
to  accomplish  this  purpose. 

CIRCLE  READINGS  FOR  INITIAL  DIRECTIONS.* 


Posi- 
tion. 

Telescope 
direct. 

Teles  :opj 
reversed. 

Posi- 
tion. 

Telescope 
direct. 

Telescope 
reversed. 

I 

0  00  40 

180  oo  40 

9 

128  OO  40 

308  oo  40 

2 

195  oi  50 

15  oi  50 

10 

323  oi  50 

143  01    50 

3 

30  03  10 

210   03    10 

II 

158  03  10 

338  03  10 

4 

225    04    20 

45  04  20 

12 

353  04  20 

173    04    20 

5 

64  oo  40 

244  oo  40 

13 

192  oo  40 

12    OO   40 

6 

259  oi  50 

79  oi  50 

14 

27  oi  50 

207  oi  50 

7 

94  03  10 

274  03  10 

15 

222    03    IO 

42  03  10 

8 

289  04  20 

109    04    20 

16 

57  04  20 

237   04    20 

For  a  method  of  correcting  azimuths  for  the  accumulated  twist 
of  triangulation,  see  page  202. 

45.  Time  for  Measuring  Horizontal  Angles. 

It  was  formerly  the  practice  to  measure  angles  only  during  that 
part  of  the  day  when  signals  appear  steady,  that  is,  during  the 
latter  part  of  the  afternoon  and  sometimes  in  the  early  morning. 
In  1902  the  Coast  Survey  parties  were  instructed  to  observe  from 
3  P.M.  until  dark,  on  heliotropes,  and  then  to  continue,  with  the 
use  of  acetylene  lights,  until  n  P.M.  The  criterion  to  be  used  in 
deciding  whether  conditions  were  favorable  was  not  the  appear- 
ance of  the  signals  themselves,  but  the  variations  of  the  measures 
of  the  angles.  The  results  showed  that  angles  can  often  be 
measured  with  sufficient  accuracy  at  times  when  the  appearance 
of  the  signals  would  indicate  poor  conditions.  From  the  results 
of  this  season's  work  it  became  evident  that  night  observations 

*  From  Coast  and  Geodetic  Survey  Special  Publication  No.  19. 


64 


FIELD-WORK  OF  TRIANGULATION 


are  somewhat  more  accurate  than  those  made  in  daylight.  Ob- 
serving at  night  is  also  more  economical  than  observing  in 
the  day  on  heliotropes,  because  at  night  the  observer  is  less 
dependent  upon  weather  conditions  (see  Art.  16). 

46.  Forms  of  Record. 

The  following  are  forms  of  record  which  may  be  used  for  hori- 
zontal angles  of  triangulation. 

HORIZONTAL  ANGLES.     DIRECTION   INSTRUMENT. 
Station,  Corey  Hill.     Date,  May  21,  1907.     Observer,  A.  N.     Recorder, 
W.  R.  N.     Inst.  No.  31.     Set  No.  2. 


Station 
observed. 

Time. 

Tele- 
scope. 

Micro. 

Circle. 

Run. 

Mean. 

Cor.  for 
run. 

Cor'd 
meas. 

0           / 

F. 

B. 

Blue  Hill 
Prospect 

h    m 
4  30 

Dir. 
Dir. 

A 
B 
C 

A 
B 
C 

15  oi 

138  30 

51-5 
54-o 
49.0 

'50-5 
53-7 
48.5 

0.6 
0-3 

51-2 
2O.  2 

Si-5 
20.9 

22.  0 

18.1 

50.9 
20.5 

21.5 
18.0 

20.3 

20.  o 

HORIZONTAL  ANGLES.     REPEATING   INSTRUMENT. 
Station,  Corey  Hill.     Date,  May  21,  1907.     Observer,  J.  N.  B.      Instr. 
B.  &  B.,  No.  1567. 


Station. 

Time. 

Tel. 

Rep. 

Ver.  A. 

B. 

Mean. 

Angle. 

Mean. 

Blue  Hill 

h    m 

3   20 

D 

0 

O   OO   OO 

/t 

OO 

00 

orn 

orn 

to 

Prospect 

P.M. 

I 

123  28  10 

20 

IS 

6 

*2o  49  40 

40 

40 

123     28     l6.7 

R 

0 

20  49  40 

40 

40 

6 

0   OO    IO 

10 

10 

123     28     15  .O 

123     28    15.8 

*  Note.  —  Since  the  angle  is  over  120  degrees  the  A  vernier  has  passed  360  degrees  twice  in  the 
six  repetitions.  In  computing  the  mean  we  divide  the  720  degrees  by  6  mentally  and  write  down 
12  — ,  then  divide  the  20  degrees  by  6,  add  the  whole  degrees  to  120,  and  then  divide  the  minutes 
and  seconds.  Observe  that  when  six  repetitions  are  used,  the  remainder,  when  dividing  the 
degrees  by  6,  gives  the  first  figure  of  the  minutes,  i.e.,  20  degrees  4-6  =  3  degrees  in  the  mean, 
plus  2  degrees  to  be  carried  to  the  minutes  column  giving  20  minutes.  Similarly  in  dividing 
the  minutes  by  6  the  remainder  is  the  tens  place  in  the  seconds. 


REDUCTION  TO  CENTER 


47.  Accuracy  Required. 

As  stated  in  paragraph  4  on  p.  60  the  degree  of  accuracy  re- 
quired on  the  Coast  Survey  triangulation  is  such  that  the  error 
of  closure  of  a  triangle  shall  seldom  exceed  3"  and  shall  average 
about  i".  The  following  list,  taken  at  random  from  a  longer 
list  in  Special  Publication  No.  19,  will  indicate  the  degree  of 
accuracy  actually  obtained  in  the  work  of  the  Coast  Survey. 


Section. 

Probable  error 
of  an  observed 

Average 
closing 
error  of  a 

Max.  cor.  to 
direction. 

Maximum 
closing 
error  of  a 

triangle. 

triangle. 

Nevada  —  California  

±0.23 

O."C7 

O.6o 

I    C7 

New  England 

rto   26 

O   7< 

I    17 

2  O2 

Eastern  Oblique  Arc. 

±0  30 

o  78 

O   74 

2    73 

Holton  Base  net.  .       

±O   34 

O   7Q 

o  84 

2    28 

Atlanta    base    to     Dauphin 

Island-base  '  

±0.36 

I  .IO 

0.84 

2    6q 

Lampasa  base  to  Seguin  base. 

±0.45 

I-I3 

1.96 

3    31 

Calif.  —  Washington  Arc  

±o-53 

I  .22 

2.03 

6-35 

48.  Reduction  to  Center. 

In  case  certain  lines  from  any  station  are  obstructed,  it  may 
become  necessary  to  set  the  instrument  over  a  point  at  one  side 
of  the  center,  called  an  eccentric  station,  and  to  measure  the 
angles  at  this  new  point. 
These  angles  are  measured 
with  the  same  degree  of  pre- 
cision as  though  the  instru- 
ment were  at  the  center. 
Before  such  angles  can  be 
used  for  solving  the  tri- 
angles, they  must  be  reduced  J 
to  the  values  they  would 
have  if  the  instrument  were  placed  at  the  center.  The  data  nec- 
essary for  the  calculation  include  the  approximate  distances  (D)  to 
the  points  sighted,  the  distance  from  the  center  mark  to  the  in- 
strument (d) ,  called  the  eccentric  distance,  and  the  angle  at  the  in- 
strument between  the  center  mark  and  each  of  the  signals  sighted. 


FIG.  34. 


66 


FIELD-WORK  OF  TRIANGULATION 


In  Fig.  34,  let  C  be  the  center,  E  the  instrument,  and  5  one  oi 
the  signals.  The  angle  CES  =  a  (called  the  azimuth),  measured 
right-handed  from  the  center  to  the  distant  signals,  may  be 
calculated  for  each  signal  by  combining  angles  already  measured, 
provided  the  line  EC  has  been  connected  with  any  one  signal  by 
means  of  an  angle.  The  angle  S  is  the  change  in  the  direction, 
or  azimuth,  of  the  triangulation  line  due  to  the  eccentricity  of  the 
instrument  station.  Solving  the  triangle  for  S,  we  have 

<y/  _    dsma    *  r 

"ZJsini"' 

It  should  be  observed  that  the  algebraic  sign  of  sin  a  shows 
whether  the  azimuth  is  to  be  increased  or  diminished. 

The  following  example  shows  the  method  employed  when 
several  angles  are  to  be  reduced  to  center  simultaneously. 

EXAMPLE  OF  REDUCTION  TO  CENTER. 
Harpers  A 

d  =  im.342  log  =  0.12755 

Eccentric  sta.  No.  i.  Colog  sin  i"  =  5.31443 

log  const.  =  5.44218 

Measured  angles:  —  Center  to  Smith's  Cupola  =  42°  14'  20",  Smith's 
Cupola  to  Cotton's  =  62°33'io".i,  Methodist  Church  to  Cotton's  = 
58°  45'  3i".o,  Cotton's  to  White  Flag  =  56°  22'  36".!,  White  Flag  to 
Baldwin's  =  43°  59'  57"-4. 

The  azimuths  from  the  center  are  computed,  and  the  computation  is 
tabulated  as  follows: 


Station. 

Smith's 
Cupola. 

Methodist 
Church. 

Cotton's. 

White  Flag. 

Baldwin's. 

Azimuth  

42°  14'  20".  o 

46°oi'59".i 

104°  47'  30".  i 

161°  10'  06".  2 

205°  io'o3".6 

Log  sin  az  
Colog  dist  
Log  const  

9.8275 
6.1052 
5.4422 

9.8572 
6.1025 
5-4422 

9.9853 
6.0640 
5-4422 

9.5090 

6  .  2672 
5.4422 

9.  6286  n 
6.0909 
5.4422 

LogS"  
S" 

1.3749 
+23"  7 

1.4019 
+25"  2 

I.49I5 
-(-31"  o 

1.2184 
+16"  5 

1.1617  n 

—  IA"    <r 

Azimuth  

42°  14'  43".  7 

46°  02'  24".  3 

104  "48'  oi".i 

161°  10'  22".  7 

205°  09'  49"-  1 

Reduced  angles:  —  Smith's  Cupola  to  Cotton's  =  62°  33'  17". 4,  Metho- 
dist Church  to  Cotton's  =  58°  45'  36". 8,  Cotton's  to  White  Flag  = 
56°  22'  21  ".6,  White  Flag  to  Baldwin's  =  43°  59'  26". 4. 

*  To  reduce  the  angle  to  seconds  we  should  divide  by  arc  i";  but  since  arc  i"  is 
nearly  equal  to  sin  i"  the  result  is  numerically  the  same  if  we  employ  the  latter. 


PHASE  OF  SIGNAL 


If  the  distances  are  not  known  with  sufficient  accuracy  at  first, 
as  might  be  the  case  where  there  are  two  eccentric  stations  in  the 
same  triangle,  it  may  be  necessary  to  obtain  the  reduced  angle 
by  a  second  approximation.  After  the  angles  have  been  reduced 
to  center,  as  already  explained,  the  lengths  of  the  lines  may  be 
calculated  with  a  greater  degree  of  accuracy  than  at  the  begin- 
ning of  the  computation.  By  using  these  improved  values  of  the 
distances  the  reduction  to  center  may  be  repeated  and  better 
values  of  the  angles  obtained. 

49.  Phase  of  Signal. 

If  the  sights  are  taken  on  pole  signals,  and  the  illumination  is 
stronger  on  one  side  than  on  the  other,  as  it  usually  would  be  in 
bright   sunlight,    the   observer 
cannot  judge   the  position  of 
the  center  but  sights  the  center 
of  the  part  that  he  can  see. 

In  this  case  it  becomes  nec- 
essary to  correct  the  observed 
angles  for  this  effect,  which  is 
known  as  the  "correction  for 
phase . "  If  the  shaded  portion 
of  the  pole  is  so  indistinct  that 
it  cannot  be  used  in  judging  the 
position  of  the  center,  then  the 
illuminated  portion  must  be 
bisected.  If  the  signal  pole  is 
cylindrical,  the  effect  of  phase 
on  the  measured  angle  may  be 
calculated  as  follows:  in  Fig. 
35,  representing  a  section  of 
the  signal  pole,  let  CS  be  a 

line  pointing  in  the  direction  of  the  sun,  and  CO  the  line  to 
the  observer.  The  limits  of  the  bright  portion  of  the  pole 
visible  to  the  observer  are  B  and  D.  By  measuring  the  angle 
between  the  sun  and  the  signal  the  observer  obtains  the  angle 


FIG.  35. 


68  FIELD-WORK  OF  TRIANGULATION 

ACS  =  a.    The  total  width  of  the  pole  is  2  r,  and  the  apparent 

width  is 

BE  =  r  +  r  cos  (180  —  a). 

The  decrease  in  width  of  the  object  is,  therefore, 

EF  =  2  r  —  r  (i  —  cos  a). 
The  angle  at  the  observing  station  subtended  by  this  distance 

is 

EF  r  (i  +  cos  «) 

J9  •  arc  i"  ~     D  •  arc  i" 

The  correction  to  the  observed  direction  is  one-half  this  amount 
since  in  each  case  the  space  is  bisected.  The  final  correction  is, 
therefore, 


50.   Measures  of  Vertical  Angles. 

The  method  of  determining  the  elevations  of  triangulation 
points  will  be  discussed  in  a  later  chapter,  but  since  the  field- 
work  of  measuring  the  vertical  angles  is  carried  on  in  connection 
with  the  measurement  of  the  horizontal  angles,  it  will  be  briefly 
discussed  here.  The  instrument  used  for  these  measurements 
may  be  a  repeating-circle  or  a  fixed  circle  read  by  microscopes. 
On  account  of  the  difficulty  of  ascertaining  the  exact  effect  of 
atmospheric  refraction,  vertical  angles  are  subject  to  much 
greater  errors  than  horizontal  angles.  A  relatively  small  num- 
ber of  measures  of  the  vertical  angle,  half  with  the  instrument 
direct  and  half  with  .it  in  the  reversed  position,  is  sufficient  to  de- 
termine the  angle  as  closely  as  the  uncertainty  of  refraction  will 
permit.  Owing  to  diurnal  changes  in  the  amount  of  the  re- 
fraction, it  is  advisable  to  make  the  measurements  between 
ii  A.M.  and  4  P.M.,  because  the  refraction  is  nearly  stationary 
during  these  hours.  About  an  hour  before  sunset  the  refraction 
is  very  uncertain.  In  recording  the  angle  it  is  essential  to  state 
exactly  the  height  of  the  instrument  above  the  station  mark  and 
also  the  exact  point  sighted,  so  that  the  angle  on  each  line  may  be 
reduced  to  that  of  the  line  between  the  two  station  marks. 


MEASURES  OF  VERTICAL   ANGLES 


Vertical  Circle. 
(Coast  and  Geodetic  Survey.) 


70  FIELD-WORK  OF  TRIANGULATION 

The  vertical  angles  may  also  be  obtained  by  means  of  the 
micrometer  in  the  eye-piece  of  the  theodolite,  if  it  is  placed  so  as 
to  measure  angles  in  the  vertical  plane.  Micrometer  readings 
on  the  different  stations,  in  connection  with  readings  of  the 
spirit  level  on  the  alidade,  will  give  the  differences  in  vertical 
angles.  If  the  vertical  angle  of  any  one  station  is  known,  the 
others  may  be  determined. 

PROBLEMS 

Problem  i.  The  circle  of  an  alt-azimuth  instrument  is  graduated  into  lo-minute 
spaces.  The  pitch  of  the  micrometer  screw  is  such  that  two  turns  are  required  to 
move  the  hairs  from  one  graduation  to  the  next.  The  head  of  the  screw  is  divided 
into  minutes  and  each  minute  into  lo-second  spaces.  The  forward  reading  (on  the 
260°  10'  line)  is  4'  03";  the  back  reading  (on  the  260°  20'  line)  is  3'  55".  What  is 
the  run  of  this  micrometer?  What  is  the  correct  reading? 

Problem  2.  The  readings  of  a  striding  level  on  a  theodolite  show  that  the  hori- 
zontal axis  is  inclined  1.5  divisions,  the  left  end  being  higher.  What  error  will  this 
cause  in  the  azimuth  reading  on  the  pole  star,  at  an  altitude  of  41°  20',  if  the  value 
of  one  division  of  the  level  is  io".o? 

Problem  3.  If  a  horizontal  angle  is  measured  between  a  mark  12°  above  the 
horizon  and  bearing  N  45°  W,  and  the  pole  star,  41°  altitude,  what  is  the  error  in 
the  angle  produced  by  an  error  of  8"  to  the  right  in  the  (collimation)  adjustment 
of  the  vertical  cross-hair. 

Problem  4.  The  angle  between  stations  A  and  B  is  measured  from  station  E 
and  found  to  be  71°  10'  i9"-5.  The  angle  from  0,  to  the  right,  to  station  A  is 
110°  15'.  The  distance  OE  is  7.460  meters.  OA  is  17,650  meters  and  OB  is  24,814 
meters.  Reduce  the  angle  to  the  center  0. 

Problem  5.  The  illuminated  portion  of  a  cylindrical  pole  is  bisected  with  the 
cross  hairs  of  a  theodolite.  The  angle  from  the  sun,  to  the  right,  to  this  signal  is 
130°  40'.  The  diameter  of  the  pole  is  6  inches.  The  distance  to  the  signal  is  8100 
meters.  What  is  the  correction  to  the  observed  direction  for  phase  of  the  signal? 


CHAPTER  IV 
ASTRONOMICAL  OBSERVATIONS 

51.  Astronomical  Observations  —  Definitions. 

In  every  trigonometric  survey,  whether  made  for  scientific 
purposes  or  for  the  purpose  of  making  maps,  it  is  essential  that 
some  of  the  triangulation  points  be  located  on  the  earth's  surface 
by  means  of  their  astronomical  coordinates.  In  determining  the 
earth's  size  and  figure  by  measuring  arcs  on  the  surface  it  is 
essential  that  the  curvature  be  determined  by  means  of  astro- 
nomical observations.  If  the  triangulation  is  used  to  control  the 
accuracy  of  a  topographical  survey,  the  astronomical  work  fur- 
nishes the  data  necessary  for  correctly  locating  and  orienting  the 
map  on  the  earth's  surface.  The  astronomical  data  also  furnish 
a  means  of  detecting  the  accumulated  twist  of  a  chain  of  triangu- 
lation, and  of  correcting  the  azimuth  at  intervals  along  the  line. 
Astronomical  observations  are  also  frequently  made  in  order  to 
supply  data  to  be  used  in  other  measurements,  as,  for  example, 
when  rating  chronometers  for  gravity  or  magnetic  observations. 
These  astronomical  observations  form  a  distinct  branch  of  geo- 
detic work. 

It  will  be  assumed  that  the  student  has  a  general  knowledge  of 
astronomy,  and  only  such  definitions  will  be  given  as  are  essential 
in  viewing  the  subject  from  the  standpoint  of  the  geodesist.  The 
astronomical  observations  which  it  is  important  for  us  to  con- 
sider include  the  determination  of  the  following  four  coordinates: 
(i)  time,  (2)  longitude,  (3)  latitude,  and  (4)  azimuth.  Before 
describing  the  instruments  and  methods,  we  will  define  the  fol- 
lowing terms  which  are  to  be  employed. 

The  vertical  at  any  point  on  the  earth's  surface  (OZ,  Fig.  36) 
is  the  direction  in  which  the  force  of  gravity  acts  at  that  point. 

71 


72  ASTRONOMICAL  OBSERVATIONS 

In  general  it  does  not  perfectly  coincide  with  the  normal  to  the 
spheroidal  surface,  and  hence  there  is  a  difference  between  the 
astronomical  coordinates  and  the  geodetic  coordinates.  The 
deflection  of  the  plumb  line  from  the  normal  at  any  place  is  called 
the  station  error.  The  point  vertically  overhead  (Z)  is  called 
the  zenith.  We  may  consider  that  the  universe  is  bounded  by  a 

z 


c 


FIG.  36.    The  Celestial  Sphere. 

sphere  of  infinite  radius,  and  that  the  zenith  is  the  point  where 
the  vertical  pierces  that  sphere.  The  horizon  (NEHS)  is  the 
great  circle  on  the  celestial  sphere  which  is  everywhere  90°  from 
the  zenith.  Its  plane  passes  through  the  observer  and  is  per- 
pendicular to  the  vertical  line.  Any  plane  which  contains  the 
vertical  line  cuts  from  the  sphere  a  vertical  circle  (HDZ). 

The  earth's  rotation  axis,  prolonged,  pierces  the  sphere  in  two 
points,  called  the  north  celestial  pole  (P)  and  the  south  celestial 
pole  (Pf).  The  great  circle  which  is  everywhere  90°  from  the 


THE   DETERMINATION   OF   TIME  73 

poles  is  the  celestial  equator  (QVR).  Any  plane  through  the 
axis  or  parallel  to  it  cuts  from  the  sphere  an  hour  circle  (PVP'). 
The  vertical  circle  which  passes  through  the  celestial  pole  is  called 
the  meridian  (SQZ) .  If  the  vertical  does  not  intersect  the  earth's 
axis,  the  meridian  plane  cannot  contain  the  axis  but  is  parallel  to 
it.  The  prime  vertical  is  a  vertical  circle  perpendicular  to  the 
meridian.  The  ecliptic  is  a  great  circle  cut  by  the  plane  of  the 
orbital  motion  of  the  earth  (MVL).  That  point  on  the  sphere 
where  the  ecliptic  and  the  equator  intersect,  and  where  the  sun 
passes  (in  March)  from  the  southern  to  the  northern  hemisphere, 
is  called  the  vernal  equinox. 

The  altitude  (h)  of  a  point  is  its  angular  distance  above  the 
horizon.  Its  zenith  distance  (f)  is  the  complement  of  the  altitude. 
The  azimuth  (Z)  of  a  point  is  the  horizontal  angle  between  the 
meridian  and  the  point.  It  is  usually  reckoned  from  the  south 
point  of  the  horizon,  right-handed,  from  o°  to  360°.  The  de- 
clination (5)  of  a  point  is  its  angular  distance  north  (+)  or  south 
(  — )  of  the  equator.  Its  polar  distance  (p)  is  the  complement  of 
the  declination.  •  The  hour  angle  (t)  of  a  point  is  the  arc  of  the 
equator  measured  from  the  meridian  westward  to  the  hour  circle 
through  the  point.  The  right  ascension  (a)^is^the  arc  of^the 
equator  measured  from  the  vernal  equinox  eastward  to  the  hour 
circle  through  the  point. 

The  astronomical  latitude  *  (0)  of  a  place  is  the  angular  dis- 
tance of  the  zenith  north  or  south  of  the  equator,  or,  in  other 
words,  the  declination  of  the  zenith.  The  longitude  (X)  of  a 
place  is  the  arc  of  the  equator  between  the  observer's  meridian 
and  a  primary  meridian,  as  Greenwich  or  Washington. 

52.   The  Determination  of  Time. 

The  determination  of  time,  practically  considered,  means  the 
determination  of  the  error  of  a  chronometer  on  the  local  sidereal 
time  at  the  station.  The  sidereal  time  (S)  at  any  instant  is  the 
hour  angle  of  the  vernal  equinox ;  it  is  usually  expressed  in  hours, 
minutes,  and  seconds.  From  a  consideration  of  the  definitions 

*  For  goedetic  latitude  see  p.  123. 


74  ASTRONOMICAL  OBSERVATIONS 

of  sidereal  time,  hour  angle,  and  right  ascension  it  is  evident  that 
the  first  equals  the  sum  of  the  other  two;  that  is, 

S  =  a  +  t.  [15] 

When  the  star  is  on  the  meridian,  t  is  obviously  equal  to  zero, 

and  we  have 

S  =  a,  [16] 

that  is,  the  right  ascension  of  any  star  is  equal  to  the  sidereal 
time  at  the  instant  when  that  star  is  passing  the  meridian.  If 
we  note  the  chronometer  reading  when  a  certain  star  is  passing 
the  meridian,  we  know  that  the  local  sidereal  time  (or  true  chro- 
nometer reading)  at  that  instant  is  the  same  as  the  right  ascension 
of  that  star  as  given  for  that  date  in  the  Ephemeris,*  and  that  the 
error  of  the  chronometer  is  the  difference  between  the  two.  The 
determination  of  time  with  a  transit  mounted  in  the  plane  of  the 
meridian  depends  upon  the  foregoing  principle. 

53.   The  Portable  Astronomical  Transit. 

The  instrument  chiefly  used  for  determining  time  and  longitude 
in  geodetic  work  is  the  portable  transit.  This  class  of  work 
necessitates  carrying  the  instrument  to  many  stations  located 
in  places  which  are  difficult  to  reach;  hence  it  should  be  light 
enough  to  be  easily  transported.  The  small  size  of  the  transit, 
however,  does  not  necessarily  imply  inferior  accuracy  in  the 
results;  it  is  found  by  experience  that  comparatively  small  in- 
struments, when  properly  handled,  give  results  of  great  accuracy. 
Indeed,  the  very  fact  that  the  instrument  is  light  is  a  point  in  its 
favor,  for  this  makes  it  easier  to  reverse,  and  obviates  certain 
difficulties  encountered  in  using  large  instruments  in  observa- 
tories, for  example,  the  error  due  to  flexure,  or  those  due  to 
temporary  strains  caused  by  reversal  of  the  instrument.  The 
portable  transit  is  usually  mounted  on  a  brick  or  concrete  pier, 
to  which  the  base  of  the  instrument  is  firmly  cemented. 

The  transit  instrument  itself  consists  of  a  telescope  with  a 

*  The  American  Ephemeris  and  Nautical  Almanac,  published  by  the  Navy 
Department. 


THE  PORTABLE  ASTRONOMICAL  TRANSIT 


75 


FIG.  37.     Portable  Transit  (with  transit  micrometer.) 
(Coast  and  Geodetic  Survey.) 


76 


ASTRONOMICAL  OBSERVATIONS 


rotation  axis  rigidly  attached  at  right  angles  to  it;  this  axis  termi- 
nates in  pivots  which  rest  in  wye  bearings  at  the  upper  ends  of  a 
pair  of  standards.  A  stride  level  is  provided  for  measuring  the 
inclination  of  the  rotation  axis.  The  axis  of  collimation,  which 
is  a  line  through  the  optical  center  of  the  objective  and  perpen- 
dicular to  the  rotation  axis,  rotates  in  a  vertical  plane  when  the 
horizontal  axis  is  truly  level.  For  the  purpose  of  determining 
the  time  the  instrument  may  be  set  in  any  vertical  plane,  for 
example,  the  vertical  plane  through  a  close  circumpolar  star;  but 
in  this  country  it  is  used  almost  exclusively  in  the  plane  of  the 
meridian. 

Fig.   37  shows  a  portable  astronomical  transit  used  for  the 
determination  of  time  and  longitude  by  the  Coast  and  Geodetic 
Survey.     The  focal  length  is  94  cm,  the  aperture  76  mm,  and 
the  magnifying  power  104  diameters. 
54.  The  Reticle. 

In  the  old  style  of  transit  the  reticle  consisted  of  several  closely 

spaced  vertical  spider  threads 
or  of  lines  ruled  on  glass,  and 
two  horizontal  threads  or  lines 
to  limit  the  portion  of  the 
vertical  threads  used  for  obser- 
vations. A  common  arrange- 
ment of  the  vertical  threads, 
when  the  chronograph  is  to  be 
used  for  recording  the  observed 
time,  is  shown  in  Fig.  38,  the 
smallest  intervals  correspond- 
ing to  about  2.5s  of  time  for  an 


FIG.  38. 

equatorial  star. 

55.   Transit  Micrometer. 

The  hand-driven  transit  micrometer  has  now  replaced  the  old 
style  of  reticle  on  the  instruments  of  the  United  States  Coast 
Survey.  In  this  instrument  (Fig.  39)  a  single  vertical  thread  is 
made  to  traverse  the  field  of  the  telescope  at  such  a  speed  that  it 


TRANSIT  MICROMETER 


77 


FIG.  39.     The  Transit  Micrometer. 
(Coast  and  Geodetic  Survey.) 


78  ASTRONOMICAL  OBSERVATIONS 

continually  bisects  the  star  that  is  being  observed.  The  record 
on  the  chronograph  of  the  passage  of  the  star  over  certain  fixed 
points  in  the  field  is  made  automatically  by  means  of  an  electric 
circuit.  An  automatic  cut-out  is  so  arranged  as  to  keep  the 
circuit  broken  except  during  four  revolutions  of  the  screw  in  the 
central  part  of  the  field.  The  contact  points  are  placed  so  as  to 
record  twenty  observations  on  the  star,  arranged  in  four  groups. 
The  observer  has  simply  to  set  the  thread  on  the  star  and  follow 
it  until  it  has  passed  beyond  the  range  of  observation.  The 
observer  does  not  know  exactly  when  the  observations  are  being 
made;  he  simply  watches  the  thread  and  the  star  and  keeps  the 
bisection  as  nearly  perfect  as  he  can.  It  is  necessary  to  use  both 
hands  in  order  to  give  the  thread  a  steady  motion.  The  result 
of  these  observations  is  the  same  as  though  the  observer  had 
noted  accurately  the  time  of  passage  of  the  star  over  20  vertical 
threads.  The  great  advantage  of  the  instrument  is  that  trie 
large  personal  error  due  to  estimating  times  of  transit  over  the 
threads  is  almost  wholly  eliminated.  A  further  advantage  is 
that  20  observations  may  be  made  in  about  ten  seconds,  on  an 
equatorial  star,  thus  permitting  •  observations  on  stars  culmi- 
nating in  quick  succession. 

56.  Illumination. 

The  field  of  the  telescope  is  illuminated  by  means  of  a  lamp. or 
an  electric  bulb  which  sends  light  through  the  hollow  axis  of  the 
instrument  to  a  mirror  at  the  center  of  the  telescope,  which  re- 
flects it  down  the  telescope  tube  to  the  reticle.  The  threads 
appear  as  black  lines  against  a  bright  field. 

57.  Chronograph. 

The  chronograph  is  a  registering  apparatus  driven  by  clock- 
work, and  connected  electrically  with  a  chronometer  and  with 
either  the  transit  micrometer  or  an  observing  key.  The  record 
is  made  on  a  sheet  of  paper  wound  around  a  drum  which  revolves 
once  per  minute.  A  pen  fastened  to  the  armature  of  an  electro- 
magnet is  carried  by  a  screw  in  a  direction  parallel  to  the  axis 
of  the  drum.  These  combined  motions  cause  the  pen  to  draw  a 


CHRONOGRAPH 


79 


line  spirally  around  the  drum.  When  the  sheet  is  laid  flat,  the 
record  appears  as  a  series  of  straight  parallel  lines.  The  chro- 
nome  ter  breaks  the  circuit  once  per  second  (or  once  per  two  seconds) , 
and  this  break  causes  the  armature  to  move  the  pen  to  one  side 
and  make  a  small  notch  on  the  record.  The  times  of  passage  of 
stars  over  the  threads  of  the  transit  are  also  recorded  in  a  similar 


FIG.  40.     Chronograph. 
(Coast  and  Geodetic  Survey.) 

manner.  The  character  of  the  two  kinds  of  marks  is  usually 
dissimilar,  and  they  may  easily  be  distinguished.  If  any  one  of 
the  chronometer  marks  on  the  record  sheet  is  identified,  then  the 
chronometer  time  of  every  mark  on  the  sheet  becomes  known, 
and  the  determination  of  the  fraction  of  a  second  for  each  obser- 
vation is  simply  a  matter  of  scaling  off  the  position  of  the  corre- 
sponding mark.  A  convenient  way  to  mark  the  time  without 
disturbing  the  sheet  is  to  make  notches  on  the  sheet  by  means 


8o 


ASTRONOMICAL  OBSERVATIONS 


f  Chronometer       ^  Condenser 

i/ii^TN 

IP        2 


V.  4 


-r   Battery 


Chronometer  Relay 


Battery  -=- 


Chronograph 


Observing  Key 


Sounder 


Battery  -^- 


^Sounder  Relay 


B  Signal  Relay 
^ 

~Z~      l_J 


Relay 

Telegrapher's'  &  Signal  Key 


Main  Line 


FIG.  41.     Electrical  Connections  —  Time  Observations  by  Key  Method. 


Chronometer  Condenser 


1 


-=?=-  Battery 


Chronometer  Relay 


Battery  -=- 


Battery 


Transit  Micrometer 


Battery 


Signal  Relay 

Telegrapher's  &  Signal  Key 


'Sounder  Relay 


Q  Telegrapl 
Ma" 

\-J~~ 


in  Line 


FIG.  42.     Electrical  Connections  —  Time  Observations  by  Transit  Micrometer 

Method. 


ADJUSTMENT  OF  THE  TRANSIT 


8l 


of  the  observing  key,  the  number  of  marks  so  made  showing  the 
number  of  some  minute  of  the  chronometer  reading.  The  speed 
and  the  diameter  of  the  cylinder  are  usually  such  as  to  make  one 
second  of  time  occupy  a  space  of  one  centimeter.  Fig.  40  shows 
a  chronograph  such  as  is  used  in  longitude  observations. 

58.   Circuits. 

The  arrangements  of  circuits  for  operating  the  chronograph 
are  shown  in  Figs.  41  and  42.  The  chronometer  is  placed  in  a 

!«  54»  56*  58s  6ft  35m  00s  2* 


12s 


18s 


22» 


-                  v                  v                  Y                  r                  \* 

W                    W                   W                   K                   Y                   +• 

\J                   v                   -J                    r                   K                    * 

+•                          K                          V                          V                         f                          \S 

J                   r                    v                    \J                   w                    K 

V                         \*                          ^                         V                         K 

t-                          K                         V                          W                          H 

V                         +.                          ~                         r                         y 

*-                          M                          \J                         K                          1^ 

\S                         \t                           v                          v                          K 

K                        K                        V-<                       I-1                        ^ 

U                              V-                               K                              V                               r 

n              K              w              K              w 

V                              —                              W-                            W                               Y~ 

>j              w               K        U  u  cminoi^T-' 

FIG.  43.     Chronograph  Record. 

separate  circuit  having  a  battery  of  only  one  cell,  in  order  to  avoid 
injury  to  the  mechanism,  and  operates  the  chronograph  circuit 
through  the  points  of  a  relay.  The  transit  micrometer  operates 
on  the  make-circuit,  which  is  converted  into  breaks  by  a  relay. 
If  a  key  is  used,  it  replaces  the  micrometer  relay  and  breaks  the 
circuit  when  the  key  is  pressed. 

Fig.  43  shows  a  portion  of  a  chronograph  record. 

59.  Adjustment  of  the  Transit. 

In  placing  the  transit  on  the  supporting  pier  before  adjusting 
it  in  the  meridian,  the  base  of  the  instrument  must  be  placed  so 


82  ASTRONOMICAL  OBSERVATIONS 

nearly  in  the  meridian  that  all  further  adjustment  in  azimuth 
may  be  made  by  the  adjusting  screws  provided  for  this  purpose. 
The  foot  plates  should  then  be  cemented  to  the  pier.  The  tele- 
scope is  focused  as  in  an  engineer's  transit  —  first  the  eye- 
piece, then  the  objective.  A  distant  terrestrial  object  may  be 
used  for  the  first  trial,  but  the  final  focusing  should  be  done  at 
night  on  the  stars.  A  difference  is  usually  noticed  between  the 
focus  required  by  day  and  that  found  at  night  when  artificial 
light  is  used. 

The  striding  level  and  the  horizontal  axis  may  be  adjusted 
simultaneously  by  placing  the  level  in  position,  reading  both 
ends  of  the  bubble,  then  reversing  it,  end  for  end,  and  taking 
another  set  of  readings.  Half  the  displacement  of  the  bubble 
may  be  corrected  by  adjustment  of  the  level  and  half  by  leveling 
the  axis. 

The  verticality  of  the  threads  or  the  micrometer  line  is  tested 
by  rotating  the  telescope  slightly  about  its  horizontal  axis  and 
noting  whether  a  fixed  object  remains  continuously  on  the  thread 
as  it  traverses  the  field  of  view.  Adjustment  is  made  by  rotat- 
ing the  diaphragm  or  the  micrometer  box  until  this  condition  is 
fulfilled. 

The  collimation  is  adjusted  by  placing  the  middle  line  of  the 
reticle  or  the  mean  position  of  the  micrometer  line  as  nearly  as 
possible  in  the  collimation  axis.  To  test  this,  point  the  wire  on 
some  object,  reverse  the  telescope  in  its  supports  (axis  end  for 
end),  and  see  if  the  object  is  still  sighted.  If  it  is  not,  bring  the 
wire  halfway  back  by  means  of  the  lateral  adjusting  screws. 

The  finder  circles  should  be  tested  to  see  if  they  read  zero  when 
the  collimation  axis  is  vertical.  Point  on  some  object,  level  the 
bubble,  and  read  the  circle.  Reverse  the  telescope,  point  on  the 
same  object,  and  repeat  the  readings.  The  mean  reading  is  the 
true  zenith  distance,  and  half  the  difference  between  the  "two 
readings  is  the  error  of  adjustment.  Set  the  vernier  to  read  the 
true  zenith  distance,  sight  the  object  again,  and  then  center  the 
bubble  by  means  of  the  adjusting  screws. 


SELECTING  THE   STARS   FOR  TIME  OBSERVATIONS          83 

To  place  the  line  of  collimation  in  the  meridian,  first  determine 
a  rough  chronometer  correction  by  leveling  the  axis  and  setting 
the  circles  for  the  zenith  distance  of  some  star  which  is  near  the 
zenith  and  which  is  about  to  culminate.  If  the  (sidereal) 
chronometer  is  nearly  regulated  to  local  sidereal  time,  the  right 
ascension  of  such  a  star  will  be  nearly  the  same  as  the  chronom- 
eter reading.  If  the  chronometer  is  not  regulated  at  all,  it  may 
be  set  approximately  right  by  calculating  the  sidereal  time  cor- 
responding to  the  mean  time  as  indicated  by  a  watch.  An 
error  of  one  or  two  minutes  will  not  cause  great  inconvenience, 
as  all  that  is  necessary  is  to  identify  the  star  and  begin  observing 
before  it  has  passed.  The  time,  at  which  this  star  will  pass  the 
middle  vertical  thread  must  necessarily  be  very  close  to  the  true 
sidereal  time  (right  ascension  of  star),  because  near  the  zenith 
the  effect  of  the  azimuth  error  on  the  observed  time  is  very  small. 
The  difference  between  the  right  ascension  of  the  star  and  the 
chronometer  reading  is  an  approximate  value  of  the  chronometer 
error.  Using  this  value  of  the  chronometer  error,  calculate  the 
chronometer  time  when  some  slowly-moving  (circumpolar)  star 
will  pass  the  meridian.  When  this  calculated  time  arrives,  point 
the  middle  thread  or  the  micrometer  thread  on  the  star,  using 
the  azimuth  adjustment  screws.  This  places  the  instrument 
nearly  in  the  meridian.  A  repetition  of  the  whole  process  (on  a 
different  pair  of  stars)  will  give  a  still  closer  approximation. 

It  is  not  necessary  or  desirable  to  spend  much  time  in  re- 
ducing the  errors  of  azimuth,  level,  and  collimation  to  very  small 
quantities.  They  should  be  so  small  as  to  cause  no  inconvenience 
in  making  the  observations  and  in  computing  the  results,  but 
since  they  must  be  determined  and  allowed  for  in  any  case,  the 
final  result  is  quite  as  accurate  if  the  errors  themselves  are  not 
extremely  small. 

60.   Selecting  the  Stars  for  Time  Observations. 

There  are  two  general  methods  of  selecting  the  stars  to  be  used 
for  a  time  determination.  The  older  method  requires  observa- 
tions on  ten  stars,  five  with  the  axis  of  the  telescope  in  one  posi- 


84  ASTRONOMICAL   OBSERVATIONS 

tion  (say  illumination  or  clamp  east)  and  five  with  the  axis  re- 
versed (illumination  or  clamp  west).  In  each  half -set  one  of  the 
stars  is  a  slow-moving  one,  that  is,  one  situated  near  the  pole. 
Of  the  remaining  four  stars  in  each  half -set  two  should  preferably 
be  north  of  the  zenith  and  two  south  of  the  zenith,  and  in  such 
positions  that  their  azimuth  errors  balance  each  other,  that  is, 
their  A  factors  (see  Art.  66)  should  add  up  to  zero. 

In  the  more  modern  method,  used  with  the  transit  micrometer, 
twelve  stars  are  employed,  six  in  each  position  of  the  axis.  None 
of  these  is  near  the  pole,  but  their  positions  are  so  chosen  as  to 
make  the  algebraic  sum  of  their  A  factors  nearly  equal  to  zero. 

By  the  older  method  the  error  in  azimuth  adjustment  is  more 
accurately  determined,  but  with  a  proper  selection  of  stars  the 
value  of  the  azimuth  correction  need  not  be  determined  so  ac- 
curately, because  it  has  a  relatively  small  effect  upon  the  com- 
puted chronometer  correction. 

In  preparing  for  observations  a  list  of  stars  should  first  be  made 
out,  giving  the  name  or  number  of  each  star,  its  magnitude,  right 
ascension,  declination,  and  zenith  distance,  together  with  the 
star  factors  depending  upon  its  position,  as  explained  later.  The 
declination  of  the  stars  chosen  should  be  such  that  the  algebraic 
sum  of  the  A  factors  is  less  than  unity.  It  is  desirable  that  the 
list  contain  as  many  stars  per  hour  as  possible,  but  sufficient 
time  must  be  allowed  for  reading  the  stride  level,  reversing  the 
instrument,  making  records,  etc.  The  telescope  should  be  re- 
versed before  each  half-set.  In  preparing  this  list  the  zenith 
distance  of  a  star  is  computed  by  the  relation 

r  =  0  -  5,  [i7] 

where  f  is  the  zenith  distance  (positive  if  south  of  the  zenith),  <£ 
is  the  latitude,  and  5  is  the  declination  (positive  for  stars  north 
of  the  equator). 

61.  Making  the  Observations. 

In  beginning  the  observations,  set  the  vernier  of  the  finding 
circle  at  the  zenith  distance  of  the  first  star  and  bring  the  bubble 


MAKING  THE   OBSERVATIONS  85 

to  the  center  of  its  scale  by  moving  the  whole  telescope.  The 
clamp  had  better  not  be  used  if  the  telescope  can  be  relied  upon 
to  remain  in  position  when  undamped.  When  the  star  appears 
in  the  field,  bring  it  between  the  two  horizontal  hairs  by  tapping 
the  telescope  with  the  finger.  Set  the  micrometer  line  on  the 
star  and  keep  it  bisected  until  the  observations  (4  turns  of  screw) 
are  completed.  If  the  instrument  is  not  provided  with  a  microm- 
eter, the  observer  simply  presses  the  observing  key  as  the  star 
passes  each  of  the  vertical  threads.  When  the  observations  are 
made  by  the  key  method,  the  observer  attempts  to  press  the  key 
as  soon  as  possible  after  the  star  is  actually  bisected  by  the  wire. 
In  doing  this  he  makes  an  error  which  tends  to  become  constant 
as  the  observer  gains  in  experience.  This  is  known  as  his  per- 
sonal equation.  Since  the  personal  equation  depends  chiefly  upon 
the  rapidity  and  uniformity  with  which  the  observer  is  able  to 
record  his  observations,  rather  than  upon  his  ability  to  bisect  the 
star's  image,  the  use  of  the  transit  micrometer  very  nearly  elimi- 
nates this  error. 

After  half  the  stars  in  one  set  have  been  observed,  the  axis 
should  be  reversed,  end  for  end,  in  the  supports.  The  striding 
level  should  be  read  one  or  more  times  during  each  half-set.  If 
the  pivots  are  not  truly  circular  in  section,  the  average  inclina- 
tion of  the  axis  may  be  found  by  taking  level  readings  with  the 
telescope  set  at  different  zenith  distances,  both  north  and  south. 

The  striding  level  should  be  used  with  great  care,  because  the 
level  corrections  may  be  relatively  large  and  cannot  be  eliminated 
by  the  method  of  observing,  as  in  case  of  the  collimation  error 
and,  to  some  extent  also,  the  azimuth  error. 

Following  is  a  record  of  a  set  of  observations  as  read  from  the 
chronograph  sheet,  together  with  the  readings  of  the  striding 
level.  (See  United  States  Coast  and  Geodetic  Survey  Special 
Publication  No.  14,  p.  21.) 


86 


ASTRONOMICAL  OBSERVATIONS 


Station,  Key  West.  Date,  Feb.  14,  1907.  Instrument,  transit  No.  2, 
with  transit  micrometer.  Observer,  J.  S.  Hill.  Recorder,  J.  S.  Hill. 
Chronometer,  Sidereal  1824. 


Star:  S.  Monocer. 

^6  Aurigae 

18  Monocer. 

f  Geminor. 

f  Geminor. 

63  Aurigae 

Clamp:  W 

W 

W 

W 

W 

W 

Level 

W 

E 

W 

E 

W 

E 

d 

d 

d 

d 

d 

d 

N62.o 

20.  o 

861.2 

19  4 

N6i.5 

19  5 

17.7 

59-5 

17.7 

59  6 

17.7 

59-7 

+44-3     -39-5 

+43-5     - 

-40.2 

+43-8 

-40.2 

+4-8 

+3-3 

+3-6 

d 

Computation  of  level  constant:  Mean  N  +  4 

20 

8  +  3.30 

+  3 

75  X( 

5-039 

=  +  0.146  = 

*w 

h  m 

h  m 

h  m 

•      h  m 

h  m 

h  m 

635 

6  39 

642 

6  46 

65 

S 

7  04 

I 

<a 
§ 

§ 

<n 

9 

cc 

CO 

C/J 

W 

m 

C/3 

32.0 

41.4 

73-4 

4L354.0 

95-3 

41-5 

50.5 

92.0 

19.530.4 

49-9 

16.2 

26.0 

42.2 

55-367.0     122.3 

32.4 

4L  I 

o.S 

41.853.5 

0.3 

41-9 

50.2 

O.I 

20.  o  30.1 

50.1 

16.5 

25.5 

2.0 

55666.5       o.i 

33  I 

40-4 

o.S 

42.852.6 

0.4 

42.5 

49-7 

0.2 

20  .  6  29  .  4 

O.O 

17.2 

24.8 

2.0 

56.465.8          0.2 

33.6 

39-8 

0.4 

43.55L9 

0.4 

43-1 

49-1 

O.2 

21.328.7 

O.O 

17-7 

24-3 

2.0 

57.I65.I          0.2 

33-9 

39-5 

0.4 

43-951-4 

0.3 

43-3 

48.8 

O.I 

21.728.3 

O.O 

18.0 

23-9 

i-9 

57.564.6       o.i 

34-6 

38.8 

0.4 

44-750.6 

0.3 

44-0 

48.1 

O.I 

22.327.6 

49-9 

18.8 

23.1 

1-9 

58.463.9       0.3 

35-0 

38-5 

o.S 

45.350.3 

0.6 

44-3 

47-9 

O.2 

22  .  8  27  .  I 

9-9 

19.1 

22.9 

2.0 

58.863.4          0.2 

35.6 

37-9 

o.S 

46.049.3 

0.3 

44.8 

47-3 

O.I 

23  .  6  26  .  4 

50.0 

19.8 

22.3 

2.1 

59.562.6       o.i 

36.1 

37-4 

o.S 

46.948.5 

0.4 

45-4 

46.6 

O.O 

24-325.7 

O.O 

20.5 

21.6 

2.1 

60.361.9          O.2 

36.4 

37-1 

o.S 

47.248.1 

0.3 

45-7 

46.3 

0.0 

24.625.4 

0.0 

20.7 

21.4 

2.1 

60.761.5          0.2 

Sum 

734-6 

Sum 

953.6 

Sum 

92I.O 

Sum 

499-8 

Sum 

420.3 

Sum  1221.9 

Mean 

36.73 

47-68 

46.05 

24-99 

21.  02 

01.10 

R* 

K 

— 

O.O2 

— 

0.03 

— 

0.02 

— 

0.02 

— 

O.O2 

—    0.02 

Bb 

+  0.14 

+ 

0.19 

+ 

0.14 

+ 

0.17 

+ 

0.16 

+  0.18 

t     6 

35 

36.85 

6    39 

47.84 

6    42 

46.17 

6    46 

25-14 

6 

58 

21.  l6 

7    05      01  .  26 

a    6 

35 

51-85 

6    40 

02.92 

6    43 

OI.2I 

6    46 

40.17 

6    58 

36.16 

7    05      16.28 

(a  —  /)      +15.00 

+15.08 

+15  04 

+15.03 

+15.00 

+15.02 

*  R,  correction  for  rate,  is  negligible  in  this  time  set. 


PIVOT   INEQUALITY  87 

62.  The  Corrections. 

The  corrections  that  have  to  be  applied  to  the  mean  of  the 
observed  times,  to  reduce  it  to  the  time  corresponding  to  the 
meridian  passage  are  those  for  (i)  level,  (2)  collimation,  (3) 
azimuth,  (4)  rate,  and  (5)  diurnal  aberration. 

63.  Level  Correction. 

The  level  correction  to  any  observed  time,  Bb,  is  made  up  of 
the  constant  b,  depending  upon  the  level  readings,  and  a  factor 
B,  depending  upon  the  position  of  the  star  and  upon  the  observer's 
latitude.  If  w  and  e  are  the  readings  of  the  west  and  east  end 
of  the  level  bubble  in  one  position,  and  wr  and  ef  the  readings  for 
the  second  position,  then  for  the  first  position,  the  inclination  of 
the  axis  of  the  level  in  terms  of  scale  divisions  is  J  (w  —  e)  ;  for 
the  second  position  it  is  ^  (it/  —  e'}  .  The  mean  of  the  two  is  the 
inclination  of  the  transit  axis,  free  from  errors  of  adjustment  of 
the  level.  If  b  represents  the  inclination,  then 

6  =  i»(w  -«)+.*(«'-«')] 
=  }  ((w  +  w')  -(e  +  e')]. 

If  d  is  the  value  of  one  division  of  the  level  scale  expressed  in 
seconds  of  arc,  then  b  in  seconds  of  time  is 


(e  +  e')],  [18] 

in  which  the  scale  divisions  are  supposed  to  be  numbered  each 
way  from  zero;  b  is  positive  if  the  west  end  of  the  axis  is  too  high. 
If,  however,  the  divisions  of  the  level  are  numbered  continuously 
from  one  end  of  the  tube  to  the  other,  the  equation  is 

b  =  jQ[(w-w')  +  (e-e')],  [19] 

in  which  the  primed  letters  refer  to  that  position  of  the  level  in 
which  the  zero  of  the  scale  is  west. 

64.   Pivot  Inequality. 

If  the  pivots  are  found  to  be  unequal  in  diameter,  then  the 
apparent  inclination  as  found  from  the  level  readings  must  be 
corrected  by  a  quantity  p,  which  is  the  inequality  as  found  by  a 


ASTRONOMICAL  OBSERVATIONS 


special  set  of  readings  of  the  level.  If  @e  and  &w  are  the  inclina- 
tions as  derived  from  the  level  readings,  and  be  and  bw  the  true 
inclinations  for  the  two  positions  of  the  axis, 


then 

also 
and 


.  He 

p   =   ~ 

be    =   Pe 


[20] 


To  determine  the  effect  of  this  inclination  error  on  the  observed 
time  of  transit  of  any  star,  let  S  (Fig.  44)  be  the  star  observed, 
and  let  HS  be  the  path  of  the  vertical  thread,  inclined  to  the  true 

N 


vertical  at  an  angle  b.  In  the  triangle  PHS  the  angle  at  P  is  the 
error  which  is  to  be  computed.  The  angle  at  H  is  b;  PS  is  the 
polar  distance,  or  90°  —  5;  HS  is  the  altitude  (nearly),  or  90°  —  £. 
From  the  triangle  PHS, 

sinP  =  smHS 

sin  H  ~  sin  PS  ' 
or  P  =  b  cos  f  sec  5  (approx.) 

=  &•£.  [21] 

The  factor  £  may  be  taken  from  Table  III  when  the  zenith 
distance  and  the  declination  of  the  star  are  known. 


AZIMUTH   CORRECTION  89 

65.  Collimation  Correction. 

The  correction  to  the  observed  time  is  cC,  c  being  the  constant 
angle  between  the  collimation  axis  and  the  mean  thread,  expressed 
in  seconds  of  time,  and  C  the  collimation  factor,  varying  with  the 
position  of  the  star.  The  colli-  PL 

mation  constant  c  may  be  found 
by  special  observations,  but  is 
usually  computed  from  the  tune 
observations  themselves,  as  ex- 
plained later;  it  is  considered 
positive  if  the  line  of  sight  is 
east  of  the  true  position  when 
the  clamp  is  east. 

In  Fig.  45,  P  is  the  pole,  S  the  star,  PN  the  meridian,  and  SL 
the  trace  of  the  thread  all  points  of  which  are  at  the  same  dis- 
tance (c)  from  PN.  The  error  is  the  angle  P.  Since  the  angle 
N  is  90°, 

.    „      sinSN      sine 

sin  P  =  ~ = > 

sin  PS      cos  5 

or  P  =  c  sec  5  =  cC.  [22] 

The  collimation  factor  C  will  be  found  in  Table  III. 

66.  Azimuth  Correction. 

The  error  of  setting  the  instrument  in  the  meridian  is  measured 
by  the  constant  a,  the  azimuth  of  the  axis  of  collimation  expressed 
in  seconds  of  time.  This  constant  is  derived  from  the  varia- 
tions in  the  observations  themselves.  In  Fig.  46,  P  is  the  pole,  Z 
the  zenith,  and  S  the  star.  In  the  triangle  PZS,  P  is  the  re- 
quired correction,  and  S'ZS  is  a,  the  azimuth  error.  Applying 
the  law  of  sines., 

sinP     _  sing 
sin  S'ZS  "cos  6' 
or  P  =  a  sin  f  sec  5 

=  a-A.  [23] 

The  azimuth  factor  A  may  be  taken  from  Table  III.     The  con- 


90  ASTRONOMICAL  OBSERVATIONS 

stant  a  is  positive  when  the  plane  of  the  axis  of  collimation  is  east 
of  south.  A  is  positive  for  all  stars  except  those  between  the 
zenith  and  the  pole. 


FIG.  46. 

67.  Rate  Correction. 

In  order  to  compute  these  corrections  it  is  necessary  to  reduce 
all  observations  of  the  chronometer  correction  to  some  definite 
epoch,  for  example,  the  mean  of  all  the  observed  times,  so  that 
variations  in  the  chronometer  correction  itself  will  not  affect  the 
determination  of  the  transit  errors.  This  is  done  by  applying 
the  correction 

R=(t-  To]  rh,  [24] 

where  /  is  the  chronometer  time  of  transit, 

To  is  the  mean  epoch  of  the  set, 

and    rh  is  the  hourly  rate  of  the  chronometer,  positive  if  losing, 
negative  if  gaining. 

68.  Diurnal  Aberration. 

The  motion  of  the  observer  due  to  the  diurnal  motion  of  the 
earth  makes  all  stars  appear  farther  east  than  they  actually  are; 
in  other  words  it  apparently  increases  their  right  ascensions.  The 
amount  of  the  correction  is  expressed  by  the  equation 

K  =  os.o2i  cos  0  sec  5.  [25] 


FORMULA  FOR  THE  CHRONOMETER  CORRECTION          91 

This  formula  may  be  derived  as  follows:  the  velocity  of  a  point 
on  the  earth's  equator  (toward  the  east)  is  0.288  mile  per  second. 
For  any  other  latitude  the  velocity  is  0.288  cos  0  mile  per  second. 
The  velocity  of  light  is  186,000  miles  per  second,  and  the  angular 


Equator 
FIG.  47- 

displacement  (*')  of  the  star  toward  the  east  point  of  the  horizon 

is  therefore  equal  to  tan"1  — — — ^.     The  effect  on  the  ob- 

186,000 

served  time  is  the  angle  K  at  the  pole,  Fig.  47.    Hence 

sin/c     _  sin/ 
sin  90°       cos  5 

or  K  =  o".3i9cos0sec6 

=  o*.o2i  cos<£  sec  6. 

Values  of  this  correction  will  be  found  in  Table  IV. 
69.  Formula  for  the  Chronometer  Correction. 
The  true  sidereal  time,  or  right  ascension  of  the  star,  is  given 
by  the  equation 

a  =  i  +  AT  +  K  +  R  +  Aa  +  Bb  +  Cc,  [26] 

in  which  /  is  the  mean  of  the  observed  transits  and  AT  is  the 
chronometer '  correction.  Since  the  corrections  for  aberration, 
rate,  and  inclination  may  be  found  directly,  they  are  applied  to 
/  at  once.  If  we  call  /i  the  value  of  /  thus  corrected,  then 

a  —  /i  =  Ar  +  Aa  +  Cc, 
or  AT  =  (a  -  /O  -  Aa  -  Cc.  [27] 


92  ASTRONOMICAL  OBSERVATIONS 

70.  Method  of  Deriving  Constants  a  and  c,  and  the  Chro- 
nometer Correction,  AT. 

The  method  shown  in  the  following  table  is  the  one  used  when 
the  observations  are  made  with  the  transit  micronometer  and 
when  the  latitude  is  less  than  50°.  For  greater  latitudes  the 
observations  are  reduced  by  the  method  of  least  squares. 


COMPUTATION  OF  TIME  SET. 

[Station,  Key  West,  Florida.     Date,  Feb.  14,  1907.     Set,  2.     Observer,  J.  S. 
Hill.     Computor,  J.  S.  Hill.] 


Star. 


Clamp. 


a  -  t. 


it. 


c. 


A. 


Cc. 


Aa. 


AT  = 
(«-«)- 
Cc-Aa. 


1.  S  Monocer... 

2.  ^5Aurigae... 

3.  18  Monocer. . 

4.  8  Geminor. 

5.  f  Geminor.. 

6.  63  Aurigae . . . 


7.  i  Geminor.. 

8.  0  Can.  Min. 

9.  a  Can.  Min. 

10.  /3  Geminor. 

11.  v  Geminor.. 

12.  0  Geminor.. 


+15.00 
+15.08 
+15.04 
+15.03 
+15-00 
+15-02 

+14-43 
+14-45 
+14-45 
+14.41 
+14.42 
+14-47 


o.oo 

+0.08 
+0.04 
+0.03 

o.oo 

+O.02 

-0.57 
-0.55 
-0.55 
-0.59 
-0.58 
-0.53 


+  I.O2 
+1.38 
+  I.OI 
+  I.2I 
+1.0? 
+  1-30 


+0.26 

-0.45 
+0.37 
—0.20 
+0.07 
-0.34 

—0.07 

+0.28 

+0.33 

—o. 

—0.19 

-0.05 


5 

+0.27 

+0.36 
+0.26 
+0.32 
+0.28 
+0.34 

-0.30 
—0.27 
—0.26 
-0.30 
-0.32 
—0.29 


+O.O2 
—0.03 
+0.03 
— O.OI 

o.oo 

— O.O2 

O.OO 
+O.OI 
+O.OI 

o.oo 
—O.oi 

0.00 


5 

+14.71 
+14.75 
+14.75 
+14.72 
+14.72 
+14.70 

+14.73 
+14.71 
+14.70 
+14.71 

+14.75 

+14.76 


1.  3 

2.  3 
5-  2 
6.  5 
9-  4 

10.  9 


.00  St  +  3. loc  +  0. 
.00  8t  +  3.89^  —  0. 
.12  S/  +  2. 75  c  —  0. 


Mean  AT  =  +  14. 727 


—  0.04  =  0 

—  0.13  =  0 

—  0.09  =  o 

—  0.13  =  o 

—  0.12  =  0 

+  2.61  =  o 


(2)  X  0.707 
(i)  +  (S) 
(6)  X  0.920 
(8)  +  (9) 


—  3. 


3. 

4.  3.00  dt  —  3.47  c  —  0.34  aE  +1.74  =  0 

7.  i 

8'.  4 

12.  - 


ii.   St  =  —  0.274  from  (10} 

5 

AT=  +  15.00  —  0.274  =  +  14.726 
1.63  =  0 


.82  «-  5.38  c 
i.  32  -5.  38  c 


+2.73  =  0 
+2.73  =  0 


(3)Xo.6o7 
(4)  +  (7) 
from  (8) 


14.    —0.82  +1.02  —0.99  aw    —0.13  =  0 
16.   —0.82—0.83+0.560^.     +1.63  =  0 


13-     c  =  +  0.262  from  (12) 

IS-  ajp  =  +0.071 
17.  aE  =  +0.036 


5 

+O.O2 
— O.O2 
—0.02 
+0.01 
+0.01 
+0.03 

O.OO 
+O.O2 
+0.03 
+0.02 
—0.02 
—0.03 


The  serial  numbers  in  the  lower  part  of  the  table  show  the 
order  of  the  different  steps  of  the  computation.     Equation  i  is 


METHOD   OF  DERIVING  CONSTANTS  93 

obtained  by  taking  the  terms  corresponding  to  the  three  southern- 
most stars  (that  is,  Nos.  i,  3,  and  5),  substituting  the  sums  of 
these  numbers  in  the  equation  AT  -j-  Cc  -f-  A  a  —  (a  —  ti)  =  o, 
and  treating  this  result  as  though  it  were  the  equation  for  a  single 
star.  Equations  2,  3,  and  4  are  found  in  a  similar  manner.  This 
gives  four  equations  for  the  twelve  stars,  two  for  each  half-set. 
Since  there  are  now  as  many  equations  as  there  are  unknowns, 
the  quantities  c,  aw,  &E,  and  AT  may  be  found  by  solving  these 
equations  simultaneously.  Notice  that  in  this  solution  15*  has 
been  dropped  from  AT,  and  that  5t  is  the  small  correction  which 
must  be  added  to  15*  to  obtain  AT. 

The  following  method  of  deriving  the  constants  and  the 
chronometer  correction  without  employing  least  squares  is 
applicable  when  the  two  groups  of  stars  have  A  factors  which  are 
not  so  nearly  balanced,  or  where  the  list  of  observed  stars  con- 
sists of  one  slowly-moving  (azimuth)  star  and  several  time  stars 
in  each  half-set.  This  method  gives,  by  a  series  of  approxima- 
tions, very  nearly  the  same  result  that  would  be  obtained  by  the 
method  of  least  squares.  The  various  steps  in  the  computation 
are  shown  in  tabular  form  in  Fig.  48. 

The  formulas  on  which  the  method  is  based  are  as  follows: 
For  each  star  we  may  write  an  equation  of  the  form 

a  -  ^  =  AT  +  Aa  +  Cc.  [28] 

Then  for  the  east  and  west  groups  we  have 

(a  -  ti)w  =  AT  +  Awaw  +  Cwc,  )'  xv 

(a  -  t,]E  =  AT  +  AwaE  +  CEc.  \ 

Assuming  at  first  that  aE  and  aw  are  equal,  we  find  an  approxi- 
mate value  of  c  by  subtracting  the  second  equation  from  the 
first.  Solving  for  c,  we  find 

—  (a  —  ti)E 


In  the  above  example, 

10.  25  —  10.17 
-  ' 


+0.03. 
1.42  +  1.34 


94 


ASTRONOMICAL  OBSERVATIONS 


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METHOD   OF  DERIVING   CONSTANTS 


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96  ASTRONOMICAL  OBSERVATIONS 

Using  this  approximate  value  of  c,  the  last  terms  in  Equations  (a) 
are  computed  and  subtracted  from  (a  —  h)  in  each  case,  leaving 
the  equations  in  the  form 

(a  -  /i  -  Ce)  =  &T  +  Awaw. 

Taking  each  half-set  separately,  and  also  grouping  the  azimuth 
star  and  the  time  stars  separately,  we  have  for  the  next  group 
(a  -  ti  -  Cc)  =  AT  +  (Awaw)  az, 
(a  -  /i  -  Cc)  =  AT  +  (Awaw)  time, 

and  a  similar  pair  of  equations  for  the  second  position  of  the  axis. 
From  Equa.  (b)  we  derive 

(a  —  ti  —  Cc)az.  —   (a  —  h   —  Cjtime 

aw=  ~  ~A~    ~A~ 

•"-az.         •**•  time 

In  the  example, 

10.08  —  10.21  0 

aE  =    ^  —  =  -0.78 

—  0.96  —  0.03 

20.74  —  19.21 

and  aw  =  — *-* —  =  —0.76. 

-2.05  +0.03 

Employing  these  approximate  values  of  aE  and  aw,  the  A  a  cor- 
rections are  computed  and  subtracted,  giving  the  value  in  the 
column  headed  a  —  t\  —  Cc  —  Aa.  For  the  time  stars  these 
values  are  19.23  and  19.19.  Since  these  values  do  not  agree  for 
the  two  positions  of  the  instrument,  the  value  of  c  is  evidently  in 
error.  A  second  approximation  must  be  made  by  treating  the 
difference  of  these  numbers  (0.04)  as  an  error  in  c  and  obtaining 
a  correction  to  c  by  the  same  process  that  was  used  in  finding  c  in 
the  first  instance,  that  is, 

r*         *•                  I9-I9  ~~  I9-23 
Correction  to  c  =  -    — z— ~  =  —0.014. 

1.42  +  1.34 
Hence  c  =  +0.03  —  0.014  =  +0.016. 

With  this  improved  value  of  c  new  values  of  a#  and  aw  are  com- 
puted as  before.  The  second  values  are  aE  =  —0.768  and  aw  = 
—0.772.  Using  these  values,  the  chronometer  corrections  are 
found  to  agree,  and  hence  no  further  approximation  is  necessary. 


DETERMINATION   OF  DIFFERENCES  IN  LONGITUDE         97 

The  azimuth  and  collimation  corrections  are  now  found  for  each 
star,  as  shown  in  the  upper  part  of  the  table.  The  mean  of  the 
AT's  for  all  the  stars  is  the  chronometer  correction  for  the  mean 
of  the  observed  times.  The  residuals  (v)  are  computed  by  sub- 
tracting AT  for  each  star  from  the  mean  of  the  AT's  for  that 
group.  These  should  add  up  nearly  to  zero. 

Whenever  the  most  accurate  results  are  desired,  the  computa- 
tion may  be  made  by  the  method  of  least  squares.  For  the 
details  of  this  method  see  Coast  and  Geodetic  Survey  Special 
Publication  No.  14,  p.  41. 

71.  Accuracy  of  Results. 

The  error  in  the  computed  value  of  AT1  due  to  accidental  errors 
alone  may  be  kept  within  a  few  hundredths  of  a  second.  Ob- 
servations made  by  the  key  method  may  be  subject  to  a  large 
constant  error,  the  observer's  personal  equation,  which  may  be 
several  times  as  large  as  the  accidental  error.  Observations 
made  with  the  transit  micrometer  are  nearly  free  from  personal 
errors. 

72.  Determination  of  Differences  in  Longitude. 

The  determination  of  the  difference  in  longitude  of  two  stations 
consists  in  measuring  the  difference  between  the  sidereal  times 
at  the  two  places.  The  method  almost  exclusively  used  for 
accurate  longitudes  in  places  where  a  telegraph  line  is  available 
is  that  in  which  the  times  are  compared  by  electric  signals  sent 
over  the  telegraph  line.  Wireless  apparatus  may  be  used  for 
this  purpose,  but  it  has  not  as  yet  come  into  general  use,  probably 
because  it  is  not  as  economical  as  the  ordinary  lines.  The 
method  used  at  present  by  the  Coast  Survey  differs  considerably 
from  the  old  method,  owing  to  the  introduction  of  the  transit 
micrometer. 

According  to  the  usual  program  each  observer,  provided  with 
transit,  chronometer,  and  chronograph,  determines  the  local 
sidereal  time  by  the  method  previously  described;  then  the  two 
chronometers  are  compared  by  means  of  arbitrary  signals,  which 
are  sent  over  the  telegraph  line  and  recorded  simultaneously  on 


98 


ASTRONOMICAL  OBSERVATIONS 


OBSERVATIONS  BY  KEY  METHOD  99 

both  chronographs;  and,  finally,  each  observer  again  determines 
the  local  sidereal  time. 

According  to  the  Coast  Survey  instructions  (Spec.  Pub.  No.  14) 
each  half -set  should  consist  of  from  5  to  7  stars  (preferably  6), 
all  of  these  to  be  time  stars  (no  azimuth  star).  The  algebraic 
sum  of  the  azimuth  factors  (A)  should  be  less  than  unity.  Four 
half-sets  are  observed  during  an  evening,  and  the  telescope  axis 
is  reversed  before  each  half-set.  The  observers  do  not  exchange 
places  during  the  occupancy  of  the  station,  as  was  formerly  the 
practice.  Observations  on  three  or  four  nights  usually  give  the 
desired  accuracy. 

Fig.  49  shows  the  switchboard  and  the  arrangement  of  the 
electric  circuits  required  in  longitude  observations.  When  the 
observer  is  making  observations  for  time,  the  circuit  is  arranged 
as  shown  in  Fig.  42. 

Fig.  50  shows  the  circuit  as  arranged  during  the  exchange  of 
arbitrary  signals.  These  signals  are  made  by  tapping  the  signal 
key  in  the  main-line  circuit.  Half  of  these  signals  are  sent  by 
the  eastern  observer,  half  by  the  western,  in  order  to  eliminate 
the  error  due  to  the  time  of  transmission  of  the  signal.  The 
chronometers  mark  the  record  sheets  while  the  signals  are  being 
sent,  so  that  the  time  of  each  signal  may  be  read  from  each 
chronograph  sheet.  The  difference  in  longitude  is  found  from 
interpolated  chronometer  corrections. 

73.   Observations  by  Key  Method. 

If  the  transit  micronometer  is  not  used,  the  selection  of  stars 
must  be  modified  so  as  to  allow  more  time  between  observations. 
Since  the  observations  will  be  subject  to  the  personal  errors  of 
both  observers,  it  is  important  that  the  observers  should  exchange 
places  at  the  middle  of  the  series,  so  that  their  relative  personal 
equation  will  enter  the  latter  half  of  the  observations  with  its 
algebraic  sign  changed.  The  arrangement  of  the  circuits  is 
shown  in  Fig.  51,  in  which  an  observing  key  replaces  the  relay  and 
circuit  of  the  transit  micrometer. 


IOO 


ASTRONOMICAL  OBSERVATIONS 


Battery 


Battery 


Signal  Relay 

Telegrapher's  &  Signal  Key 
Main  Line 


FIG.  50.     Electrical  Connections  —  Exchange  of  Signals  —  Transit 
Micrometer  Method. 


Battery 


Signal  Relay 

Telegrapher's  &  Signal  Key 
Main  Line 


FIG.  51.    Electrical  Connections  —  Exchange  of  Signals  — Key  Method. 


DETERMINATION  OF   LATITUDE  COI 

74.  Correction  for  Variation  of  the  Pole. 

The  periodic  variation  of  the  position  of  the  pole  affects  all 
observations  for  longitude  and  must  be  allowed  for  by  applying 
the  corrections  given  in  tables  published  annually  by  the  Inter- 
national Geodetic  Association.  (See  Art.  81,  p.  106.) 

75.  Determination  of  Latitude. 

The  method  which  has  been  chiefly  used  in  this  country  for 
determining  astronomical  latitudes  for  geodetic  purposes  is  that 
known  as  Talcott's  (or  the  Harrebow-Talcott)  Method.  The 
instrument  employed  is  the  zenith  telescope,  illustrated  in  Fig.  52. 
The  principle  involved  is  that  of  measuring,  not  the  absolute 
zenith  distances  of  stars,  as  is  done  with  the  meridian  circle, 
but  the  small  difference  between  the  zenith  distances  of  two  stars 
which  are  on  opposite  sides  of  the  zenith.  By  a  proper  selection 
of  stars  this  difference  in  zenith  distance  may  be  made  so  small 
that  the  whole  angular  distance  to  be  measured  comes  within 
the  range  of  the  eye-piece  micrometer,  which  for  most  instru- 
ments is  about  half  a  degree.  A  sensitive  spirit  level  attached 
to  the  telescope  serves  to  measure  any  slight  change  in  the  in- 
clination of  the  vertical  axis  of  the  instrument  between  the  two 
observations  on  a  pair  of  stars.  The  accuracy  of  the  results 
obtained  by  this  method  is  superior  to  that  of  every  other  field 
method,  and  compares  favorably  with  the  results  obtained  with 
the  largest  instruments. 

The  horizontal  axis  of  the  telescope  is  very  short  as  compared 
with  that  of  the  transit  instrument;  small  errors  in  the  inclination 
of  the  axis,  however,  have  very  little  effect  upon  the  results;  a 
close  adjustment  is  therefore  unnecessary.  Since  the  instrument 
is  used  in  the  plane  of  the  meridian  and  must  be  quickly  turned 
from  the  north  side  to  the  south,  or  vice  versa,  the  horizontal 
circle  is  provided  with  stops  which  are  adjustable,  so  that  the 
telescope  may  be  quickly  changed  from  one  side  of  the  zenith  to 
the  other.  The  micrometer,  placed  in  the  focal  plane  of  the  eye- 
piece, is  set  so  as  to  permit  of  measuring  small  angles  in  the  verti- 
cal plane.  The  head  of  the  screw  is  graduated  to  read  to  about 


102 


:ASTRONOMICAL  OBSERVATIONS 


FIG.  52.    Zenith  Telescope. 
(Coast  and  Geodetic  Survey.) 


SELECTING  STARS  103 

o".5  directly  and  to  o".o5  by  estimation.    The  spirit  level  has  an 
angular  value  of  one  (2wm)  division  equal  to  about  i".5. 

76.  Adjustments  of  the  Zenith  Telescope. 

When  the  instrument  is  in  perfect  adjustment,  the  plate  levels 
should  be  central  in  all  azimuths  as  the  telescope  is  turned 
about  the  vertical  axis.  The  leveling  may  be  perfected  by  use 
of  the  more  sensitive  latitude  level.  The  horizontal  axis  must 
be  at  right  angles  to  the  vertical  axis.  The  movable  micrometer 
threads  must  be  truly  horizontal.  They  may  be  adjusted  by  a 
method  similar  to  that  used  in  adjusting  the  engineer's  level  — 
by  swinging  the  telescope  horizontally  through  a  small  angle  and 
observing  whether  the  thread  remains  on  a  fixed  point.  The 
collimation  adjustment  should  be  made  in  the  same  manner  as  in 
a  transit,  but  is  not  of  so  great  importance.  Allowance  must  be 
made  for  the  eccentricity  of  the  telescope  when  making  the  colli- 
mation adjustment.  The  value  of  one  turn  of  the  micrometer 
may  be  determined  approximately  by  observations  upon  a  close 
circumpolar  star  near  its  elongation.  The  most  satisfactory  way, 
however,  is  to  derive  the  value  of  one  turn  from  the  latitude 
observations  themselves,  by  the  method  of  least  squares.  The 
value  of  one  division  of  the  latitude  level  may  be  determined  by 
means  of  a  level  trier,  or  it  may  be  found  by  varying  the  inclina- 
tion of  the  telescope  and  employing  the  eye-piece  micrometer  to 
determine  the  amount  of  this  inclination  by  observations  on  a 
terrestrial  mark. 

When  in  use  the  instrument  is  mounted  on  a  wooden  or  con- 
crete pier.  It  is  usually  protected  by  a  tent  or  other  temporary 
shelter. 

In  order  to  make  the  observations,  it  is  necessary  to  have  a 
chronometer  regulated  to  local  sidereal  time  with  an  error  not 
exceeding  one  second  of  time. 

77.  Selecting  Stars. 

The  list  of  stars  in  the  American  Ephemeris  will  not  ordinarily 
be  sufficient  for  latitude  observations,  on  account  of  the  exacting 
nature  of  the  conditions.  It  will  be  necessary  to  consult  such 


IO4 


ASTRONOMICAL  OBSERVATIONS 


star  catalogues  as  Boss's  Preliminary  General  Catalogue  of  6188 
stars  for  the  Epoch  1900,  or  one  of  the  Greenwich  catalogues.  In 
order  to  keep  the  zenith  distances  within  the  required  limits,  it 
will  often  be  necessary  to  observe  on  stars  which  are  much  fainter 
than  those  used  for  time  observations.  The  pairs  of  stars  selected 
should,  if  possible,  differ  by  less  than  2om  in  their  right  ascension 
and  by  less  than  20'  in  their  declinations.  The  actual  zenith 
distance  of  a  star  should  not  exceed  45°.  Following  is  a  speci- 
men star  list  for  zenith  telescope  observations. 


OBSERVING  LIST   (FORM   i). 

[St.  Anne,  111.,  June   25,   1908.     Zenith  telescope  No.   4. 
Search  factor  =  2  <j>  =  82°  03'.] 


41°  01^.3. 


0) 

d 

, 

0 

6  A 

•   • 

A    . 

*.. 

Jo 

M 

rt 

•^  9 

"SI 

^J    CH 

ll> 

g-|g 

-o- 

<N 

I« 

1* 

||> 

in 
B 

"3 

S 

i.| 

h^  -^  *° 

II    "U.S 

I 

i  ' 

IH    S 

•^3  <u^_ 

3 

&  g 

*§ 

S9 

p 

^•s" 

W 

fe;3 

1° 

1^3° 

H 

Ami 

0             , 

0             , 

0             / 

' 

0             / 

4327 

4-5 

16  55  22 

82  II 

N 

12 

4379 

4-9 

17  ii  53 

—  O    21 

82     32 

81  50 

-13 

-17 

S 

41  16 

28 

4441 

5-9 

17  28  13 

28    28 

S 

10 

4494 

5-8 

17  42  04 

53  5o 

25    22 

82  18 

+  15 

+  20 

N 

12    41 

30 

4623 

5-i 

l8    13    22 

64    22 

N 

24 

4651 

5-4 

18  18  45 

17  47 

46  35 

82  09 

+  6 

+  8 

S 

23  18 

16 

4669 

5-9 

l8    22    26 

29  47 

•s 

20 

4711 

5-5 

18  31  52 

52  17 

22    30 

82  04 

+  i 

+  I 

N 

ii  15 

20 

*  a  =  number  of  turns  of  the  micrometer  screw  in  one  minute  of  arc 
one  turn  of  the  micrometer  screw  =  44".6so. 


1.34.    The  value  of 


78.  Making  the  Observations. 

In  observing  on  a  pair  the  finder  circle  is  set  for  the  mean  of  the 
two  zenith  distances,  and  the  level  is  brought  nearly  to  the  center 
of  the  tube.  If  the  northerly  star  of  the  pair  culminates  first, 
the  telescope  is  set  on  the  north  side  of  the  meridian  by  means 
of  the  azimuth  stop.  When  the  star  enters  the  field,  the  observer 
bisects  it  with  the  micrometer  line.  If  a  pair  of  lines  is  used,  the 
star  is  centered  in  the  space  between  the  two.  When  the  star  is 
on  the  meridian,  as  shown  by  the  chronometer  reading,  the  bi- 


FORMULA  FOR  THE  LATITUDE 


105 


section  of  the  star  is  perfected;  the  latitude  level  is  read  im- 
mediately, and  then  the  scale  of  the  micrometer  screw.  As  soon 
as  these  readings  are  recorded,  the  telescope  is  turned  to  the  south 
side  of  the  meridian  and  the  bubble  is  brought  to  the  center,  if 
necessary,  by  moving  the  whole  telescope.  In  leveling  the 
bubble  the  tangent  screw  of  the  setting  circle  must  not  be  dis- 
turbed in  any  case,  because  the  accuracy  of  the  method  depends 
upon  preserving  a  fixed  relation  between  the  direction  of  the 
zero  micrometer  reading  and  the  axis  of  the  latitude  level.  The 
slightest  change  in  the  angle  between  these  two  during  the  ob- 
servations on  a  pair  will  render  the  observations  worthless. 
When  the  southern  star  appears  in  the  field  the  pointing  and  the 
readings  are  made  exactly  as  for  the  northern  star. 

79.  Formula  for  the  Latitude. 

The  principle  involved  in  this  method  may  be  seen  in  Fig.  53. 
The  latitude,  EZ,  as  derived  from  the  southern  star,  is 

EZ  =  ES,  +  StZj 

or  <f>  =  5,  +  f,, 

and  from  the  northern  star  it  is 

EZ    =   ESn   -   ZSn, 

or  d>  =  8n  —  fn. 


FIG.  S3- 


The  mean  of  the  two  values  of  <j>  is 


106  ASTRONOMICAL  OBSERVATIONS 

If  we  let 

na  and  s8  =  the  level  readings  for  the  southerly  star, 
nn  and  sn  =  the  level  readings  for  the  northerly  star, 
d  =  the  angular  value  of  one  division  of  the  level, 
r8  and  rn  =  the  refraction  corrections, 
M8  and  M  n  =  the  micrometer  readings, 

and         R  =  the  value  of  one  turn  of  the  micrometer,  then  the 

latitude  is  determined  by  the  equation 


-2  (M,  -  Mn)  -R  +  -  {(n,  +  O  -  (s. 


[30 


This  formula  applies  .when  the  zero  of  the  level  scale  is  in  the 
center  of  the  tube.  If  the  zero  is  at  the  eye-piece  end  of  the 
tube,  the  level  correction  is 

»  7 

+  -  {  (n,  -  nn)  +  (s,  -  sn)}. 
4 

If  for  any  reason  the  observations  are  not  made  when  the  star  is 
exactly  on  the  meridian,  another  term  must  be  added  to  the  above 
formula;  this  will  be  of  the  form  -f-  f  (ma  +  mn)  when  m,  and  mn 
are  the  reductions  of  the  measured  zenith  distances  to  the  true 
zenith  distances.  (See  Special  Publication  No.  14,  p.  119.)  For 
the  application  of  least  squares  to  the  computation  of  latitude  see 
Chauvenet,  Spherical  and  Practical  Astronomy  ;  Hay  ford,  Geodetic 
Astronomy;  and  Coast  and  Geodetic  Survey  Special  Publication 
No.  14. 

80.  Calculation  of  the  Decimations. 

When  the  stars  selected  are  not  found  in  the  Ephemeris,  it 
will  be  necessary  to  calculate  the  apparent  declinations  for  the 
date  of  the  observation.  Formulae  and  tables  for  making  these 
reductions  will  be  found  in  Part  II  of  the  Ephemeris.  See  also 
Coast  and  Geodetic  Survey  Special  Publication  No.  14,  p.  116. 

81.  Correction  for  Variation  of  the  Pole. 

The  observed  latitude  may  be  in  error  by  several  tenths  of  a 
second,  owing  to  the  fact  that  the  observed  value  necessarily 


REDUCTION   OF  THE   LATITUDE   TO   SEA-LEVEL 


107 


refers  to  the  position  of  the  pole  at  the  date  of  the  observation, 
whereas  the  fixed  value  of  the  latitude  of  a  place  is  that  referred 
to  the  mean  position  of  the  pole.  Fig.  54  shows  the  plotted 
positions  of  the  pole  for  every  o.i  year  during  the  period  1900.0 
to  1906.0  (Jordan).  The  coordinates  of  the  instantaneous  pole 


0.30 


o.io 


0.00 


0.10 


0.20 


0.30 


-0.20 


SCALE  OF   FEET 


0  10  20  30 

FIG.  54.    Motion  of  the  North  Pole,  1900  to  1906. 

and  data  for  correcting  observed  values  are  published  annually 
by  the  International  Geodetic  Association,  and  observations  may 
be  referred  to  the  mean  pole  by  employing  these  tables. 

82.  Reduction  of  the  Latitude  to  Sea-Level. 

In  order  that  all  latitudes  may  refer  to  the  same  level  surface, 
they  are  all  reduced  to  their  values  at  sea-level.  If  we  suppose  a 


io8 


ASTRONOMICAL  OBSERVATIONS 


lake  surface,  in  the  northern  hemisphere,  to  be  at  a  great  height 
above  sea-level,  then  it  may  be  shown  that  the  northern  end  of 
this  lake  surface  is  actually  nearer  to  the  surface  of  the  sea  than 
is  the  southern  end  of  the  lake  surface.  If  we  imagine  a  series 
of  such  surfaces  at  varying  heights  above  sea-level,  it  is  obvious 
that  the  vertical  is  a  curved  line,  since  it  must  at  every  point  be 
normal  to  the  level  surface  passing  through  that  point.  Evi- 
dently this  curved  line  is  concave  toward  the  earth's  rotation 
axis.  To  correct  an  observed  latitude  at  elevation  h  to  the  cor- 
responding latitude  at  sea-level,  it  is  necessary  to  apply  the 
correction 

A<£  =  —  o".o52  h  sin  2  <£, 


where  h  is  in  thousands  of  feet, 
formula  becomes 


If  h  is  expressed  in  meters,  the 


—0.000171  h  sin  2 


(See  Art.  170,  p.  256.)  Values  of  this  correction  will  be  found  in 
Table  VII.  Below  is  an  example  of  the  form  of  record  and  com- 
putation of  latitude  from  Special  Publication  No.  14.) 


RECORD  OF  LATITUDE  OBSERVATION. 

[Station,  St.  Anne.     Date,  June  25,  1908.      Chronometer,  2637.      Observer, 

W.  Bowie.] 


Star 

Micrometer. 

Level. 

Chro- 
nom- 

Chro- 

Meri- 

No. of 
pair. 

number 
Boss 

Nor 
S. 

eter 
time  of 

nometer 
time  of 
observa- 

dian 
dis- 

Re- 
marks. 

cat. 

Turns. 

Div's. 

North. 

South. 

culmi- 

tion. 

tance. 

nation. 

42  .6 

4623 

N 

24 

88.2 

9-2 

71.6 

103.8 

18  13  18 

II 

4651 

S 

16 

66.0 

42.2 

8-7 

* 

18  18  39 

* 

+  i6t 

103.2 

71.0 

*  These  columns  used  only  when  star  is  observed  off  the  meridian. 

t  This  is  the  continuous  sum,  up  to  this  pair,  of  the  south  minus  the  north  micrometer  turns. 


ACCURACY   OF  THE   OBSERVED   LATITUDE 
LATITUDE  COMPUTATION 


109 


Date. 

Catalogue. 

Micrometer. 

Level. 

Merid- 
ian dis- 
tance. 

Declination. 

Star 
No. 

Nor 
S. 

Reading. 

Diff.  Z.  D. 

N. 

S. 

Diff. 

1908. 

June  25 

4623 
4651 

N 
S 

24  88.2 
16  66.  c 

*.        d. 
—8  22.2 

09.2 
7I.6 
42.2 
103.2 

42.6 
103.8 
08.7 
71.0 

d. 
-1.05 

S. 

64   21   59.53 
17  46  48.62 

Sum  and  half 
sum. 

Corrections. 

Latitude. 

Remarks. 

Micrometer. 

Level. 

Refrac- 
tion. 

Meridian. 

o         /            // 

82  08  48.15 
41   04   24.08 

/        a 
[-3  03-56 

a 
-o-39 

—  O.o6 

41   Ol    20.07 

Value  of  one  division  of  latitude  level:  Upper  —  i".6co 

Lower  —1.364 

Mean  —1.482 
Value  of  one  turn  of  micrometer  =  44". 650 

83.  Accuracy  of  the  Observed  Latitude. 

The  latitude  may  be  determined  by  this  method  with  a  prob- 
able error  of  from  o".3  to  0^.4  from  one  pair  of  stars.  The  final 
value  for  the  latitude  of  the  station  determined  from  as  many 
pairs  of  stars  as  can  be  observed  on  one  night  may  be  found  with 
an  error  of  from  0^.05  to  o".io  (or  5  to  10  feet).  It  is  not  consid- 
ered advisable  to  observe  the  same  pair  of  stars  on  several 
nights,  as  was  formerly  the  practice,  owing  to  the  comparatively 
large  errors  in  the  declinations  themselves.  The  present  practice 
is  to  observe  each  pair  but  once  and  to  observe  such  a  number 
of  pairs  that  the  uncertainty  of  the  final  latitude  is  not  greater 
than  o".io. 

In  view  of  the  fact  that  nearly  every  latitude  is  affected  by  a 
station  error  which  may  amount  to  several  seconds,  and  that  the 
real  object  of  the  observation  is  to  determine  this  station  error, 
it  is  better  to  determine  a  large  number  of  latitudes  with  the 
degree  of  accuracy  above  mentioned  than  to  attempt  to  diminish 


HO  ASTRONOMICAL  OBSERVATIONS 

the  error  of  observation  and  occupy  but  a  small  number  of 
stations.  This  results  in  the  practice  of  occupying  stations  but 
one  night,  unless  for  some  reason  it  is  apparent  that  the  required 
accuracy  will  not  be  reached  without  additional  observations. 

84.   Determination  of  Azimuth. 

When  determining  an  azimuth  for  the  purpose  of  orienting  a 
triangulation  system,  the  observer  usually  has  a  choice  of  several 
methods,  all  of  them  capable  of  yielding  the  required  accuracy, 
for  example,  (i)  measuring  the  angles  between  a  circumpolar 
star  and  the  triangulation  lines. by  means  of  the  direction  in- 
strument, (2)  measuring  from  a  triangulation  station  to  a  cir- 
cumpolar star  with  the  repeating  instrument,  or  (3)  measuring 
from  a  circumpolar  star  to  an  azimuth  mark  with  the  micrometer 
of  a  transit  instrument.  In  all  determinations  of  azimuth  it  is 
necessary  to  know  the  local  time  in  order  to  compute  the  azimuth 
of  the  star.  This  must  be  found  by  special  observations,  unless, 
as  is  often  the  case,  the  longitude  is  being  determined  at  the  same 
time  and  the  chronometer  correction  is  already  known.  For  the 
purpose  of  orienting  the  primary  triangulation  it  is  necessary  to 
determine  the  azimuth  with  an  error  -not  exceeding  o".$o.  At 
Laplace  stations  (coincident  triangulation,  longitude,  and  azi- 
muth stations),  where  the  accumulated  twist  of  the  chain  of  tri- 
angles is  to  be  determined,  it  is  desirable  to  determine  the  azi- 
muth within  o".3o  or  less.  It  is  also  desirable  that  the  instru- 
ment station  and  the  azimuth  mark  should  both  be  triangulation 
stations.  When  horizontal  angles  are  being  measured  at  night, 
the  azimuth  observation  is  made  a  part  of  the  same  program  by 
including  pointings  on  a  circumpolar  star  with  the -regular  series 
of  pointings  on  lights  at  the  triangulation  stations.  An  azimuth 
found  by  this  method  is  more  accurate  than  one  determined  by 
means  of  an  auxiliary  point  and  subsequently  connected  with 
the  triangulation  by  means  of  a  horizontal  angle  measured  by 
daylight. 

On  account  of  the  slow  apparent  motions  of  stars  near  the  pole, 
nearly  all  accurate  azimuth  observations  are  made  on  close  cir- 


FORMULA   FOR   AZIMUTH 


III 


cumpolars,  since  errors  of  the  latitude  and  the  time  have  less 
effect  on  the  result  than  for  stars  farther  from  the  pole.  The 
stars  ordinarily  used  for  azimuth  observations  are  shown  in 
Fig.  55- 


*   51  Cephei 


Xll' 


FIG.  55.     Circumpolar  Stars. 

85.  Formula  for  Azimuth. 

In  general  all  these  methods  consist  in  calculating  the  azimuth 
of  the  star  at  the  instant  of  observation  and  combining  this 
azimuth  with  the  measured  horizontal  angle  from  the  star  to  the 
station.  The  azimuth  of  a  circumpolar  star  is  found  by  the 
formula 

,,  sinJ  r     T 

tanZ  = : >  [33] 

cos  <j>  tan  5  —  sin  <j>  cos  t 

where  Z  is  the  azimuth  measured  from  the  north  toward  the  east, 
and  /  is  the  hour  angle. 

If  Equa.  [33]  be  divided  by  cos  <£  tan  5,  then 


tanZ  =  — 


cot  6  sec  <£  sin  t 


i  —  cot  5  tan  <f>  cos  t 

=  —  cot  6  sec  <j>  sin  /  ( -     - )  • 
\i  —  a/ 


[34] 


112  ASTRONOMICAL  OBSERVATIONS 

If  values  of  —  -  —  are  tabulated  *  this  formula  will  be  found  more 
i  —  a 

convenient  than  Equa.  [33]. 

86.  Curvature  Correction. 

In  computing  the  azimuth  of  the  star  it  would  evidently  be 
inconvenient  to  apply  the  formula  to  each  separate  pointing  on 
the  star,  on  account  of  the  large  amount  of  computation.  It  is 
simpler  and  sufficiently  accurate  to  calculate  the  azimuth  of  the 
star  at  the  mean  of  the  observed  times  of  pointing,  and  then  to 
correct  the  computed  azimuth  for  the  small  difference  between 
this  azimuth  and  the  mean  of  all  the  azimuths.  The  correction 
for  this  difference  is 


T   ..  »j  o 

Curvature  Correction  =  —  tanZ-  z\  —  —  —  >          he] 

n  **  sin  i 

in  which  n  =  the  number  of  pointings 

and  r  =  the  interval  of  time  (in  seconds)  between  the  ob- 

served time  and  the  mean. 

The  sign  of  the  correction  is  such  that  it  always  decreases  the 
angle  between  the  star  and  the  pole.  For  the  derivation  of  this 
formula  see  Hayford's  Geodetic  Astronomy,  p.  213.  The  correc- 

tion may  also  be  written  in  the  form  —tan  Z  [6.73672]  -  V  T2. 

(See  Doolittle's  Practical  Astronomy,  p.  537.) 

87.   Correction  for  Diurnal  Aberration. 

On  account  of  the  motion  of  the  observer,  due  to  the  earth's 
rotation,  the  star  is  apparently  displaced  toward  the  east.  The 
correction  to  the  computed  azimuth  for  the  effect  of  this  apparent 
displacement  is  given  by  the  expression 

^        f      AI  //      cosZcos<6  r  ,, 

Corr.  for  Aberra.  =  o  .32-  •  [36] 

COS  n 

This  correction  is  always  positive  for  an  azimuth  counted  clock- 
wise. For  the  derivation  of  this  formula  see  Doolittle's  Practical 
Astronomy,  p.  530. 

*  For  a  table  of  values  of  log  -  see  Special  Pub.  No.  14. 


THE  DIRECTION  METHOD  113 

88.  Level  Correction. 

If  the  horizontal  axis  is  not  level  when  a  pointing  is  made  on  the 
star,  the  observed  direction  must  be  corrected  by  the  following 
quantity: 

Lev.  Corr.  =-[(w  +  w')  -  (e  +  e')]  tan h.  [37] 

4 

For  proof  of  this  formula  see  pp.  55  and  87.  If  the  level  is 
graduated  from  one  end  to  the  other, 

Lev.  corr.  =  -  [(w  -  wr)  +  (e  -  e'}}  tan  h,  [38] 

4 

where  w  and  e  are  read  before,  and  w'  and  e'  are  read  after,  the 
reversal  of  the  striding  level.  If  the  azimuth  mark  is  not  near 
the  horizon,  it  is  necessary  to  apply  a  similar  correction  to  the 
observed  direction  of  the  mark.  The  correction  is  to  be  added 
algebraically  to  readings  which  increase  in  a  clockwise  direction. 

89.  The  Direction  Method. 

In  observing  for  azimuth  by  this  method  the  observations  are 
carried  out  almost  exactly  as  in  measuring  the  angles  of  a  tri- 
angulation,  except  that  the  chronometer  is  read  whenever  a  point- 
ing is  made  on  the  star,  and  level  readings  to  determine  the 
inclination  of  the  axis  are  made  just  before  or  just  after  pointing 
on  the  star.  The  altitude  of  the  star  should  be  measured  at  least 
twice  during  the  observations.  In  observing  Polaris  in  connec- 
tion with  a  number  of  triangulation  stations,  it  is  best  to  take 
the  pointing  on  the  star  last.  From  twelve  to  sixteen  sets  should 
be  made  with  the  direction  instrument,  in  order  to  secure  the 
necessary  precision. 

Following  is  an  example  of  the  form  of  record  and  computation 
of  an  azimuth  by  the  method  of  directions. 


114 


ASTRONOMICAL  OBSERVATIONS 


HORIZONTAL  DIRECTIONS 

[Station,  Sears,  Tex.  (Triangulation  Station).     Observer,  W.  Bowie.     In- 
strument, Theodolite  168.     Date,  Dec.  22,  1908.] 


Position. 

Objects 
observed. 

d 

6 
H 

Q* 

g° 

JJ 

S 

Backward. 

6*52 

£g 

g 
a 

Jj 

a§ 

&c 

S3  0 

Q+3 

Remarks. 

h  m 

" 

i 

Morrison  .  . 

8  19 

D 

A 

o 

o 

35 

35 

i  division  of  the 

5 

4i 

41 

striding   level 

C 

36 

34 

37.0 

=  4"-  194 

R 

A 

180 

oo 

36 

35 

B 

32 

31 

C 

35 

34 

33.8 

35-4 

oo.o 

Buzzard... 

D 

A 

53 

30 

43 

42 

B 

41 

42 

C 

34 

33 

39  2 

R 

A 

233 

30 

39 

37 

B 

34 

32 

C 

38 

38 

36.3 

37-8 

02.4 

Allen  

D 

A 

170 

14 

61 

62 

B 

57 

55 

C 

61 

59 

59-2 

R 

A 

350 

14 

50 

49 

B 

63 

60 

C 

53 

53 

54-7 

57-0 

21.6 

Polaris.... 

D 

A 

252 

01 

54 

53 

W                   E 

h  m    s 

B 

54 

53 

9.3               28.0 

i  48  35-5 

C 

51 

51 

52-7 

27-7                9-i 

i  51  06.0 



18  4        o  ^  18  Q 

i  49  50.8 

R 

A 

72 

OI 

09. 

09 

24-9                6-3 

B 

02 

OI 

13-0               31-7 

C 

10 

08 

06.5 

29.6 



H.  9  -13-5  25.4 

-  7.0 

THE  DIRECTION  METHOD 


COMPUTATION   OF  AZIMUTH,   DIRECTION  METHOD. 

[Station,  Sears,  Tex.     Chronometer,  sidereal  1769.    <f>  =  32°  33'  31" 
Instrument,  theodolite  168.     Observer,  W.  Bowie.] 


Date  1908,  position 

Dec.  22,  i 

2 

3 

Chronometer  reading 

I    49     50  8 

2    oi      33  o 

2    16     31  o 

2    43       28  8 

Chronometer  correction  

—     4     37  .  5 

—      4     37  5 

—      4     37  4 

Sidereal  time  
a  of  Polaris  

i    45      13  3 
I   26     41  .  9 

i    56     55-5 
i    26     41.9 

2    II      53.6 
i    26     41  8 

2    38      51-5 
I    26     41  8 

t  of  Polaris  (time)  

o   18     31.4 

o   30     13.6 

o    45      ii.  8 

i    12     09  7 

t  of  Polaris  (arc) 

4°  37'  5i"  o 

7°  33'  24"  o 

11°  17'  57"  o 

18°  02'  25"  5 

5  of  Polaris                    .              ... 

88    49     27  4 

log  cot  &  

8  .  31224 

8  31224 

8  31224 

8  31224 

log  tan  <f>  

9.80517 

9.80517 

9  80517 

9  80517 

log  cos  t 

9  99858 

9  99621 

9  99150 

0  078ll 

log  a  (to  five  places)  

8.11599 

8.11362 

8.10891 

8  09552 

log  cot  S 

8  312243 

8  312243 

8  312243 

log  sec  <t> 

o  074254 

o  074254 

o  074254 

log  sin  /  .  .            ... 

8.907064 

9  118948 

9  292105 

94QOQ24 

log-i-.. 

0.005710 

0.005679 

o  005618 

o  005445 

i  —  a 

log  (-tan  A)  (to  6  places)  
A  =  Azimuth  of  Polaris,  from  north* 
Difference   in   time  between   D. 
andR...   . 

7.299271 
o   06     50.8 
m     s 

2     3O 

7-5III24 
o   ii     09.2 
m    s 

2     CO 

7.684220 
o   16     36.9 
m    5 
3    18 

7.882866 

o   26       15.0 
m     s 
i    "*8 

Curvature  correction 

O 

o 

o 

o 

Altitude  of  Polaris  =  h 

o                t         II 

33        46 

33        46 

33        46 

ri      46 

d 

-  tan  h  =  level  factor  

0.701 

o  701 

0.701 

o  70! 

4 
Inclination  "J"  

—  7  ° 

—  7  2 

—  7  o 

—I  8 

Level  correction  

—4  9 

—  5  o 

—  4  9 

—  I  3 

Circle  reads  on  Polaris  

252        01  29  6 

86        58  ii  2 

281        54  27  o 

116      45  48  6 

Corrected  reading  on  Polaris  
Circle  reads  on  mark  

252        01  24.7 
170        14  57  o 

86       5806.2 
5        15  58  2 

281        54  22.1 

200        17  42  4 

116      45  47.3. 
35      18  45  4 

Difference,  mark  —  Polaris 

278        13  32  3 

278        17  52  o 

Corrected  azimuth  of  Polaris,  from 
north  *  

o        06  50  8 

O          II   09  2 

o       16  36  9 

o      26  15  o 

180       oo  oo.o 

180       oo  oo.o 

180       oo  oo.o 

180      CO  oo.o 

Azimuth  of  Allen  .  . 

98       06  41  5 

98        06  42  8 

08         06  43  4 

08       06  4"?  I 

(Clockwise  from  South) 

To  the  mean  result  from  the  above  computation  must  be  applied  corrections  for  diurnal  aberra- 
tion and  eccentricity  (if  any)  of  Mark. 

Carry  times  and  angles  to  tenths  of  seconds  only. 

*  Minus,  if  west  of  north. 

t  The  values  shown  in  this  line  are  actually  four  times  the  inclination  of  the  horizontal  axis 
in  terms  of  level  divisions. 


n6 


ASTRONOMICAL  OBSERVATIONS 


90.   Method  of  Repetition. 

In  observing  by  the  repetition  method  the  program  given  on 
p.  57  is  followed,  with  the  addition  of  readings  of  the  chronom- 
eter and  the  stride  level,  taken  when  the  telescope  is  pointing  at 
the  star.  The  altitude  of  the  star  should  be  measured,  if  possible, 
but  may  be  computed  from  the  known  time  if  necessary.  The 
verniers  are  read  only  at  the  beginning  and  end  of  a  half  set,  as 
when  measuring  the  angles  of  a  triangulation. 

Following  is  an  example  of  the  form  of  record  and  computation 
of  an  azimuth  by  the  method  of  repetition. 

RECORD  —  AZIMUTH  BY  REPETITIONS. 

[Station,  Kahatchee  A.     State,  Alabama.     Date,  June  6,  1898.     Observer, 
O.  B.  F.     Instrument,  lo-inch  Gambey  No.  63.     Star,  Polaris.] 

[One  division  striding  level  =  2.  "67.] 


Objects. 

Chr.  time 
.on  star. 

3 

•o 

1 

1 

1 

0> 

rt 

Level  read- 
ings. 
W.        E. 

Circle  readings. 

Angle. 

• 

' 

A 

B 

S 

h    m     s 

at            it 

Mark 

D 

o 

178 

O3 

22    ^ 

2O 

21  .  2 

Star  

14.  4.6   3O 

I 

4?    IO   7 

i  /O 

wo 

•*^  •  0 

J-T"     tj.\J     £\s 

.  ^       J.W  .   f 

9-2     5-9 

49  08 

2 

52  Si 

D 

3 

9.6    5.6 

5-2  i?.o 

56  10 

R 

4 

11.3     4.0 

7-8     7.4 

Set  No.  5.. 

14  59  12 

5 

IS  oi  55 

R 

6 

8.7     6.6 

IOO 

16 

2O 

2O 

2O 

72  57  50.2 

ii.  9     3-4 

14  54  17-7 

68.2  53.6 

+  14-6 

Star  

IH   O4  44 

R 

i 

II    Q       3    A 

X  3      V/£f.      if  if 

•*•••••  y        O  *  H1 

8.5     6.8 

07  18 

2 

09  54 

R 

3 

7-9     7-3 

II.  2      4.1 

Set  No.  6.. 

14  IS 

D 

4 

9.0    6.1 

5-9    9-6 

16  14 

5 

15  18  24 

6 

5-9    9-6 

9.1     6.2 

Mark  

D 

177 

27 

oo 

OO 

OO 

72  51  46.7 

15  ii  48.2 

69-4  53-i 

+  16.3 

METHOD   OF  REPETITION 


117 


COMPUTATION  —  AZIMUTH  BY  REPETITIONS 

[Kahatchee,  Ala.    <f>  =  33°  13'  40" '.33.] 


Date   1898  set 

June  6                «; 

June  6              6 

Chronometer  reading 

14      «C4      17    7 

15     ii     48  2 

Chronometer  correction         

—  311 

—  31    I 

Sidereal  time                     

14     53     46.6 

iq      ii      17    I 

a  of  Polaris           

I       21       2O  .  3 

I      21      2O  3 

t  of  Polaris  (time) 

I"?       32        26    3 

1  3      40       l6    8 

/  of  Polaris  (arc) 
6  of  Polaris                                  .... 

203°  06'  34".  5 
88    41:     46  q 

207°  29'  12".  o 

log  cot  5                    

8  33430 

8.33430 

lop  tan  <b 

981620 

9  81629 

log  cos  /  

.oxu^y 

9.96367n 

9.94798w 

log  a  (to  five  places)  

8  .  ii426n 

8  0985771 

log  cot  5 

8    3343OS 

8    3343O(» 

log  sec  <f> 

O  O77S3S 

O  O77<3<? 

log  sin  t                        

950383OH 

9  66421  in 

i 

9QQ4387 

9004^84 

g  i-a  

log  (  —  tan  A}  (to  6  places)  

8  00005771 

8  07063571 

A  =  Azimuth    of    Polaris,    from 
north  * 

o°  34'  22"  8 

o°  40'  26"  8 

rand2SinH 

m 

7  47.7  119.3 

5  09-7     52.3 
i  26.7      4.1 

m      s 

7  04.2     98.1 
4  30.2     39.8 
i  54-2       7-1 

1    sini"    ' 
Sum 

i  52.3       6.9 
4  54.3     47.2 
7  37.3  114.0 

343   8 

2    26.8       II.  8 

4  25.8     38-5 
6  35-8     85.4 
280   7 

Mean                        .... 

C7      7 

46  8 

iog  i  y  2sip2^ 

i  7«;8 

I   670 

8w^    sini" 
log  (curvature  coir.) 

q  7  eg 

9741 

Curvature  correction  

-0.6 

-0.6 

Altitude  of  Polaris  =  h  

32°  07' 

-  tan  h  =  level  factor.  . 

O  4IQ 

o  410 

4 
Inclination   . 

4-3  6 

4-4   I 

Level  correction. 

-i"  <? 

—  I       7 

Angle,  star  —  mark  .. 

72      «C7      «JO   2 

72      *>I      4.6   7 

Corrected  angle.  ..    ... 

72      "?7      48   7 

72      <JI      45.O 

Corrected  azimuth  of  star  *  

0      34       22.2 

0      40       26  .  2 

Azimuth  of  mark  E  of  N 

73       32       IO   Q 

73      32      II  .2 

Azimuth  of  mark  

180    oo    oo.o 

2C3      32      IO  Q 

180    oo    oo.o 

2<J3       32       II  .2 

(Clockwise  from  south) 

To  the  mean  result  from  the  above  computation  must  be  applied  corrections  for  diurnal  aberra- 
tion and  eccentricity  (if  any)  of  Mark.    Carry  times  and  angles  to  tenths  of  seconds  only. 
*  Minus  if  west  of  north. 


1*8  ASTRONOMICAL  OBSERVATIONS 

91.  Micrometric  Method. 

In  employing  this  method  it  is  necessary  to  place  a  mark  nearly 
in  the  same  vertical  plane  with  the  star  at  the  time  of  the  obser- 
vation. For  greatest  accuracy,  as  well  as  for  convenience,  the 
star  should  be  observed  when  near  its  greatest  elongation.  Near 
culmination  the  star's  motion  will  carry  it  beyond  the  range  of 
the  micrometer  in  a  comparatively  short  time.  The  small 
difference  in  azimuth  between  the  star  and  the  mark  is  to  be 
measured  with  the  micrometer  in  the  eyepiece  of  a  transit  in- 
strument. The  instrument  is  clamped  in  azimuth,  and  the  read- 
ings are  taken  in  the  following  order:  take  five  pointings  on  the 
mark;  point  toward  the  star  and  place  the  stride  level  in  position; 
take  three  pointings  on  the  star  with  their  corresponding  chro- 
nometer times;  read  and  reverse  the  stride  level;  take  two  more 
pointings  on  the  star,  noting  the  times;  read  the  stride  level; 
reverse  the  horizontal  axis  of  the  instrument  in  the  bearings, 
point  the  telescope  at  the  star,  and  place  the  level  in  position; 
take  three  pointings  on  the  star,  with  chronometer  times;  read 
the  level  and  reverse  it;  take  two  more  pointings  on  the  star  and 
the  times;  read  the  level;  finally,  take  five  pointings  on  the  mark. 
Three  such  sets  will  be  found  to  require  from  thirty  to  fifty 
minutes'  time.'  Either  the  altitude  or  the  zenith  distance  of  the 
star  should  be  read  twice  during  the  set,  in  order  that  an  altitude 
for  use  in  calculating  the  azimuth  may  be  interpolated. 

The  angle  given  by  the  micrometer  readings  is  in  the  plane  of 
the  line  of  collimation  and  the  horizontal  axis.  To  reduce  this 
angle  to  the  horizontal  plane,  multiply  it  by  the  secant  of  the 
altitude.  Each  half -set  may  be  reduced  separately.  The  alti- 
tude for  the  middle  of  each  half  set  may  be  used  for  reducing  to 
horizontal.  The  value  of  one  turn  of  the  micrometer  screw  may 
be  found  by  observing  a  circumpolar  star  near  culmination,  or, 
better  still,  by  measuring  a  small  angle  by  means  of  a  theodolite 
and  then  measuring  this  angle  with  the  micrometer. 

Following  is  an  example  of  record  and  computation. 


MICROMETRIC  METHOD 


119 


RECORD  AND  COMPUTATION —  AZIMUTH   BY  MICROMETRIC 

METHOD 

[Station  No.  10,  Mexican  Boundary.    Date,  Oct.  13, 1892.    Observer,  J.  F.  H.  Instrument,  Fauth 
Repeating  Theodolite,  No.  725  (10  in.).    Star,  Polaris  near  eastern  elongation.] 


Cir- 
cle. 

Level  readings. 
W           E 

Chronom- 
eter time. 

T. 

2  sin1  \  T 

Micrometer  read- 
ings— 

sin  i" 

On  star. 

On  mark. 

km     s 

m    s 

E 

8.0       9.9 

9  06  38.0 

358.6 

31.05 

i8'.379 

i8e.3io 

\  =  2h  I2~  W  Of 

10.          7-3 

07  32.0 

3  04.6 

18.59 

0.388 

6.315 

Washington 

+18.0  —17.2 

08  05.5 

2  31.  i 

12.45 

0.400 

0.315 

0  —  31    19'  35" 
i   div.   of  level 

+08 

09  13.0 

i  23.6 

3-82 

0.424 

0.311 

=  3".  68 

E 

09  48.0 

o  48.6 

1.29 

0.430 

0.316 

i  turn  of  mic. 

18.4042 

18.3134 

Means 

W 

9.0       9.0 

9  12  01.8 

I  25.2 

3.96 

18.100 

18.290 

7.0      10.9 

12  24.7 

1  48.1 

6-37 

O.IOO 

0.275 

+16.0  -19.9 

12  48.3 

2   II.  7 

9-46 

0.090 

0.279 

-3-9 

13  36  3 

2  59-7 

17.61 

0.086 

0.281 

W 

Mean       i*.ss 

13  58.1 

3  21.  S 

22.14 

0.080 

0.279 

9  10  36.6 

12.67 

18.0912 

18.2808 

Means 

f  of  star  at  middle  of  first  half  of  set  =  58°  48'.         cosec  f  =  i .  1691.    cot  58°  47'  =  0.606. 
f  of  star  at  middle  of  second  half  of  set  =  58°  46'.    cosec  f  =  1.1695. 
o  =  i*  2Cm  07M-  S  -  88°  44'  io".4- 

ColUmati9n  axis  reads  i  (18.3134  +  18.2808)  *  =  18* .  2971 
*  In  this  instrument  increased  readings  of  the  micrometer  correspond  to  a  movement  of  the 

line  of  sight  toward  the  east  when  the  vertical  circle  is  to  the  east,  and  toward  the  west  when  the 
vertical  circle  is  to  the  west. 

Mark  east  of  collimation  axis  18.3134  —  18.2971  =    o  .0163  =   02". 02 

Circle  E.,  star  E  of  collimation  axis  (18.4042  —  18.2971)  (1.1691)  =    o  .1252 
Circle  W.,  star  E  of  collimation  axis  (18.2971  —  18.0912)  (1.1695)  =    o  .2408 

Mean,  star  E  of  collimation  axis  =    o  .  1830  =    22  .64 

Mark  west  of  star  =    20  . 62 

Level  correction  (1.55)  (0.92)  (0.606)  =  — o  .86 

Mark  west  of  star,  corrected  =    19  .76 
Mean  chronometer  time  of  observation  =    21*  10**   36* .  6 

Chronometer  correction  =  —  211    28  .2 

Sidereal  time  =     18  59   08  .4 

a  =       i    20   07  .4 

Hour-angle,  /,  in  time  17    39   01  .o 

"              in  arc  264°  45'  I5".o 
=     8.34362 
=     9.78436 
=     8.96io8n 

=  7.o89o6« 

=  8.343618 

=  0.068431 

=  9 .998177 n 


log  cot  5 
log  tan0 
log  cos  t 
log  a 
log  cot  5 
log  sec  0 
log  sin  t 


9.999467 


log  (-tan  A) 

A 

log.  12.67 
log.  curvature  corr. 


-0.33 
+0.32 


i  .  10278 


=     9  51247 


Curvature  corr. 
Diur.  Aber.  corr. 
.  91          Mean  azimuth  of  star 

Mark  west  of  star  

Azimuth  of  mark,  E  of  N  =  +1°  2/  57".  14 


=  +i°28/i6" 
19 


.8. 


120  ASTRONOMICAL  OBSERVATIONS 

92.  Reduction  to  Sea-Level. 

If  the  azimuth  mark  is  at  a  high  elevation,  the  computed 
azimuth  must  be  reduced  to  its  value  at  the  point  where  the 
vertical  through  the  mark  intersects  the  sea-level.  This  cor- 
rection in  seconds  is 


in  which  h  is  the  elevation,  0  is  the  latitude,  a  is  the  azimuth,  and 
e  and  a  are  for  the  Clarke  Spheroid  of  1866  (see  Art.  102^ 
p.  136).  If  h  is  expressed  in  meters,  this  becomes 

-f  o".oooio9  h  cos2  <£  sin  2  a.  [40] 

(log  of  0.000109  =  6.0392  —  10.) 

If  the  mark  is  either  northeast  or  southwest  of  the  observing 
station  the  observed  azimuth  must  be  increased  to  obtain  the 
correct  azimuth;  if  the  mark  is  northwest  or  southeast,  the  ob- 
served azimuth  must  be  decreased. 

Reduction  to  Mean  Position  of  the  Pole. 

The  observed  azimuth  must  be  reduced  to  its  value  correspond- 
ing to  the  mean  position  of  the  pole.  In  latitude  50°  (northern 
United  States)  this  correction  may  be  as  great  as  half  a  second 
(see  p.  107). 

PROBLEMS 

Problem  i.  What  should  be  the  linear  distance  between  the  vertical  threads  of 
a  transit  having  a.3o-inch  focus  in  order  to  give  28.5  intervals  of  time  between 
threads  for  an  equatorial  star? 

Problem  2.  The  following  readings  were  taken  to  determine  the  pivot  inequality 
of  a  transit.  Clamp  east,  level  direct,  w  =  43.5,  e  =  34.0;  level  reversed,  w  =  36.7, 
e  =  41.0.  Clamp  west,  level  direct,  w  =  39.1,  e  =  37.0;  level  reversed,  w  =  34.2, 
e  =  41.8.  The  value  of  one  division  of  the  level  is  o".75.  This  level  has  the  zero 
at  the  center  and  is  numbered  both  ways.  Find  the  pivot  inequality. 

If  a  star  is  observed  with  the  transit  in  the  position  clamp  east  what  is  the  level 
correction  to  the  observed  time  of  transit  if  6  =  +30°  and  $  =  +40°? 

Problem  3.  If  the  collimation  axis  of  a  transit  has  a  true  bearing  of  S  o°  oo'  15"  E 
what  is  the  correction  to  the  observed  time  of  transit  of  a  star  if  5  =  -f-  39°  and 


Problem  4.    If  a  latitude  is  found  to  be  36°  49'  50".  261  at  an  altitude  of  6250  feet 
what  will  this  latitude  be  when  reduced  to  sea-level? 


PROBLEMS  121 

Problem  5.  Compute  the  latitude  from  the  following  zenith  telescope  observa- 
tions. 

Star  No.  2125,  south;  chr.  time  13*  37"*;  micrometer  i6'.oo3;  level,  n  83.0,  s  30.0. 
Star  No.  2141,  north;  chr.  time,  13*  43"*;  micrometer,  13*.  504;  level  n  31.0,  s  83.5. 
Eyepiece  on  side  toward  micrometer  head;  level  zero  on  side  opposite  to  eyepiece. 
Declination  of  2125,  28°  34'  oo".8o;  declination  of  2141,  39°  oo' 08.80.  One 
division  of  latitude  level  =  i".oo.  One  turn  of  micrometer  =  2'.559. 


CHAPTER  V 
PROPERTIES  OF  THE  SPHEROID 

93.   Mathematical  Figure  of  the  Earth. 

In  calculating  the  positions  of  survey  points  on  the  earth,  it  is 
necessary  to  consider  these  points  as  lying  upon  some  mathe- 
matical surface,  like  the  sphere  or  the  ellipsoid,  taken  to  repre- 
sent the  figure  of  the  earth.  This  is  accomplished  by  projecting 
the  position  of  the  station  vertically  downward  onto  the  surface 
in  question.  The  actual  shape  of  the  earth's  surface  is  quite 
irregular  and,  from  the  nature  of  .the  problem,  can  only  be  de- 
termined approximately.  But  even  if  it  could  be  found  exactly, 
it  would  not  be  adapted  to  the  purpose  of  computation.  For 
this  reason  it  is  necessary  to  select  some  figure,  the  use  of  which 
will  simplify  the  computation,  but  which  will  nowhere  depart 
from  the  true  figure  by  an  amount  sufficient  to  produce  serious 
errors  in  the  results.  The  figure  generally  adopted  is  the  oblate 
spheroid  or  ellipsoid  of  revolution.  Such  a  figure  is  generated  by 
rotating  an  ellipse  about  its  shorter  axis.  This  surface  ap- 
proaches much  nearer  the  actual  figure  of  the  earth  than  does  the 
sphere,  but  perhaps  not  quite  so  near  as  an  ellipsoid  of  three  un- 
equal dimensions.  The  latter,  however,  would  be  an  incon- 
venient figure  to  use,  and  the  gain  in  accuracy  would  be  very 
slight. 

The  oblate  spheroid  is  an  ellipsoidal  surface  with  two  of  its 
axes  equal,  but  with  the  third  axis,  about  which  the  figure  rotates, 
shorter  than  the  other  two.  All  plane  sections  of  such  a  surface 
are  ellipses,  except  those  cut  by  planes  perpendicular  to  the  rota- 
tion axis.  Sections  through  the  rotation,  or  polar,  axis  are 
ellipses  whose  major  axes  are  the  equatorial  diameter,  and  whose 
minor  axes  are  the  polar  diameter,  of  the  spheroid.  The  nature 

122 


PROPERTIES  OP  THE  ELLIPSE 


123 


of  this  surface  will  be  understood  best  if  we  investigate  first  the 
properties  of  the  ellipse  which  generates  the  spheroid. 

94.  Properties  of  the  Ellipse. 

In  Fig.  56,  PP'  is  the  polar  axis  of  the  spheroid,  and  EE'  is  any 
one  of  the  equatorial  diameters.  F  is  one  focus  of  the  ellipse. 
At  M,  any  point  on  the  curve,  the  line  MA  is  drawn  tangent  to 
the  ellipse;  MH  is  perpendicular  to  the  tangent,  that  is,  normal 


FIG.  56. 


to  the  curve.  MH  is  the  direction  that  the  plumb  line  at  M  is 
supposed  to  assume  unless  deflected  by  local  causes,  such  as 
variations  in  density.  The  distance  MH  (  =  ^V),  terminating  in 
the  minor  axis,  is  called  the  normal.  MD  (  =  n)  is  the  normal 
terminating  in  the  major  axis.  The  angle  made  by  the  normal 
with  OE',  that  is,  with  the  plane  of  the  earth's  equator,  is  the 
geodetic  latitude  (0).*  The  angle  made  by  MO  with  OE'  is  the 
geocentric  latitude  (\f/). 

Another  angle  which  is  of  importance  in  the  geometry  of  the 
ellipse  is  the  eccentric  angle,  or  reduced  latitude,  0.  It  is  the  angle 
E'Om,  Fig.  57,  in  which  M  is  any  point  on  the  ellipse,  MN  is 

*  The  astronomical  latitude  is  the  angle  made  by  the  actual  direction  of  gravity 
(plumb  line)  with  the  plane  of  the  equator. 


124 


PROPERTIES  OF  THE  SPHEROID 


perpendicular  to  OE',  and  m  is  the  point  where  this  perpendicular 
cuts  the  circle  whose  center  is  O  and  radius  OE'. 

The  equation  of  the  ellipse  whose  major  and  minor  semiaxes 
are  a  and  b,  referred  to  its  own  axes  as  coordinate  axes,  is 


i. 


To  determine  the  coordinates  of  any  point  M  (Fig.  56),  in  terms 
of  the  latitude,  differentiate  this  equation  and  the  result  is 


a2  dy 


FIG.  57. 


Since  the  tangent  line  to  an  ellipse  makes  an  angle  with  the  axis 


of  X  whose  tangent  is 


.   dy 

1C   — £ 


dx' 
tan  (90° 


or 


dx 

-- 

dy 


The  eccentricity  e  is  the  distance  from  the  focus  to  the  center 


RADIUS  OF  CURVATURE  OF  THE  MERIDIAN  125 

divided  by  a,  that  is  -r-= .    From  the  triangle  OFF  it  will  be  seen 
OE 

that  ^  =  ~~^ — »  \ 

or  -;  =  i  -  e2. 

a2 

Therefore  (i)  may  be  written 

(*) 


From  the  equation  of  the  ellipse, 

Squaring  (2)  and  substituting  in  the  result  the  value  of  y5  from 
(3),  we  obtain* 

a  cos  <f>  r    , 

x  =  —=====  [41] 


a  (i  —  e2  )  sin  </>  r    , 

and  y  =     )         /.     "  •  [42] 

vi  -  ^sm2 


95.  Radius  of  Curvature  of  the  Meridian. 
To  find  the  radius  of  curvature  of  the  meridian  (7O>  aPpty  the 
general  formula 


dx? 


dy          x  & 
/  =  ---  - 
^          3;  a2 

*  The  relation  i  +  tan2  <f>  =  sec2  <f>  is  used  in  this  transformation. 


T^         /  \ 
From  (i) 

^          3;  a2 


126  PROPERTIES  OF  THE  SPHEROID 

Differentiating  this  equation,  we  have 

d\ 


dx2 


VI     ,  x2  b*\ 
=  -  I  -v  4-  —  •  —  i 

a2/  V  r  y    a2/ 


JL 

a2/ 


} 

Therefore       Rm  =  - 


[a4/  + 

I 


i  —  eL  sin"5  <£         i  —  eL  sm*<f> 


64a2cos2<fr  1 
i  —  e2  sin20  J 


Then,  since  b2  =  a2  (i  -  e2), 

a  (i  -  e2)      „  r    , 

~(i-e2sin2^)f 

Values  of  log  Rm  will  be  found  in  Table  X. 

96.  Radius  of  Curvature  in  the  Prime  Vertical. 

The  radius  of  curvature  of  the  surface  of  the  spheroid  in  a  plane 
at  right  angles  to  the  meridian  may  be  proved  to  be  equal  to  the 
length  of  the  normal  (N)  terminating  in  the  minor  axis.  If  a 
central  section  be  taken  through  a  point  M  and  perpendicular  to 
the  meridian,  and  the  radius  of  curvature  of  this  ellipse  at  point 


,,  ,  j    .,      .„  ,      ,  —  <> 

M  be  computed,  it  will  be  found  to  be  p  =  —  i  -  *  — 

cos  <f>  sec  \f/ 

According  to  Meunier's  theorem  the  radius  of  curvature  of  the 
normal  section  equals  the  radius  of  curvature  of  this  central 
section  divided  by  the  cosine  of  the  angle  between  the  two  planes, 

*  The  negative  sign  indicates  only  the  direction  of  bending.    It  is  customary  to 
regard  the  value  of  Rm  as  positive. 


RADIUS  OF  CURVATURE  IN  THE  PRIME  VERTICAL   127 

that  is,  by  cos  (<£  —  t).    Hence  the  radius  of  curvature  of  the 
prime  vertical  section  is  N. 

To  show  this  geometrically,  let  A  and  B  in  Fig.  58  be  two  points 
on  the  same  parallel  of  latitude.  The  normals  to  the  surface  at* 
A  and  B  always  intersect  at  H  on  the  minor  axis.  Let  C  be  a 
point  on  the  prime  vertical  section  through  A,  and  also  on  the 
meridian  of  B.  The  normals  at  A  and  C  intersect  at  some  point 
K  above  H.  K  is  approximately  the  center  of  curvature  of  the 


FIG.  58. 

arc  AC.  When  the  meridian  PBC  is  taken  nearer  to  A,  points 
A  and  C  approach  each  other,  the  intersection  of  their  normals 
approaches  the  true  center  of  curvature,  and  the  length  CK 
approaches  the  true  radius  of  curvature.  But  the  nearer  C 
approaches  J.,  the  nearer  it  approaches  B.  Hence  CK  must 
ultimately  coincide  with  AH;  that  is,  H  is  the  point  toward  which 
the  center  of  curvature  is  approaching  and  the  normal  N  is  the 
radius  of  curvature  of  the  prime  vertical  section  at  A . 
From  Fig.  56  it  is  evident  that 

N=-^ 


[44] 


Values  of  log  N  will  be  found  in  Table  X. 


128  PROPERTIES  OF  THE  SPHEROID 

^^ 
The  normal  terminating  in  the  teai*q)  axis  is 


n  = 


r     -, 
[45] 


, 
Vi-e2sin2</> 

The  radius  of  the  parallel  of  latitude  (  =  x)  is  given  by 

Rp  =  N  cos  <£.  [46] 

97.  Radius  of  Curvature  of  Normal  Section  in  any  Azimuth. 

Having  found  the  radii  of  curvature  of  the  two  principal  sections, 
it  now  remains  to  find  a  general  expression  for  the  radius  of  cur- 
vature in  any  azimuth,  and  it  will  be  shown  that  this  may  be 
expressed  in  terms  of  the  two  radii  already  found. 
The  equation  of  the  spheroid  is 


or  bV  +  b  V  +  flV  =  a?b\  (a) 

In  Fig.  59  the  Zi-axis  coincides  with  the  polar  axis  of  the  spheroid. 
If  M  be  any  point  on  the  meridian  ZiM,  and  MY  any  section  cut 
by  a  plane  through  MH  (the  normal)  making  an  angle  a  with  the 
meridian,  then  the  equation  of  the  spheroid  may  be  transformed 
so  as  to  refer  to  the  origin  C  and  the  new  Z  axis  CM.  Let  the 
coordinates  of  any  point  P  be  #1,  y\t  z\,  and  let  the  new  coordi- 
nates be  x,  y,  z.  Then,  from  Fig.  59,  the  relation  of  the  new, 
coordinates  to  the  old  is  given  by 

xi  =  OG  =  OC  +  x  +  2  cos  <f>  +  y  cos  a  sin  <j> 
=  Ne*  cos  <f>  +  x  +  2  cos  0  -f  y  cos  a  sin  <£, 

yi  =  y  sin  a, 

Zi  =  2  sin  0  —  y  cos  a  cos  0. 

Substituting  these  values  in  (a), 

b2  (Ne*  cos  0  -f  s  +  2  cos  0  +  3;  cos  a  sin  <£)2  +  #y  sin2  « 

-f  a2  (s  sin  <£  —  y  cos  «  cos  0)2  =  a2Z>2, 

which  is  the  equation  of  the  spheroid  referred  to  the  new  axes. 
If  x  is  made  equal  to  zero,  then  P  will  be  on  the  curve  M  Y,  and 
the  equation  becomes  the  equation  of  this  plane  section,  that  is, 


RADIUS  OF  CURVATURE  OF  NORMAL  SECTION 


129 


b2  (Ne*  cos  0  +  z  cos  <£  +  y  cos  a  sin  <f>)2  +  #y  sin2  a 

+  a2  (z  sin  0  —  y  cos  a  cos  </>)2 

the  equation  of  the  ellipse  MY. 


FIG.  59. 

To  determine  the  radius  of  curvature  at  M  it  is  necessary  to 

find  -7-  and  -  -  and  to  substitute  these  values  in  the  general 
dy          dy* 

formula  for  radius  of  curvature. 

Expanding  the  last  equation,  collecting  terms,  and  dividing 
through  by  a2, 
y2  [i  -  #  (i  -  cos2  a  cos2  0)]  +  z2  (i  -  e2  cos2  0) 

—  yz  (i  #  cos  a  sin  <f>  cos  #)  +  2  y  (i  —  e2)  •  #  •  e2  cos  a  sin  </>  cos  <f> 

+  22e2(i  -  e2)  .^.cos2^  =  (i  -  e2)  (a2  - 


130  PROPERTIES  OF  THE  SPHEROID 

or,  in  abbreviated  form, 

y*A  +  z*B  -  yzC  +  2yD  +  2zE  =  F. 

Differentiating  this  equation,  y  being  taken  as  the  independent 
variable, 


dy  dy  dy 

Differentiating  again, 


75 
2B 

y 


A/£\2  ^dz 
(-r)  -  2C  — 
\dl  </ 


dy2  2  Bz  -  Cy  +  E 

For  point  M  ,  y  =  o  and  z  =  n  =  N  (i  —  e2).    Therefore 
dz  = 
dy~ 

N(i—  e2)  (2  e*  cos  a  sin  <f>  cos  cf>)  —  2  (i  —  e2)Ne2  cos  a  sin  <ft  cos  <ft 
2Bz-Cy  +  E 

2  [i   -  g2  (i   -  COS2  a  COS2  </))] 


,  =      _ 

~ 


i  —  e2  -f  g2  cos2  a  cos2  </> 


N  (i  -  e2) 
_  (i  —  e2)  (sin2  a  +  cos2  a)  +  e2  cos2  a.  (i  —  sin 

N(i-  e2) 
(i  —  e2)  sin2  a  +  cos2  «  —  cos2  a  •  e2  sin2  0 


sin2  a  -\ ^-r  cos2  a  (i  —  e2  sin2 

i  —  e2 


in2o:  +  N  CQ&  a 

NRm 


THE  MEAN  VALUE  OF  £«  131 

Substituting  these  differential  coefficients  in  the  usual  formula 
for  radius  of  curvature,  we  have 

r.,.1 
L47J 


N  cos2  a  +  Rm  sin2  a 
If  a  =  o°, 


the  radius  of  curvature  of  the  meridian;  and  if 


a  =  90°, 


then  R^  =        =  =  N, 

Rm 

the  radius  of  curvature  of  the  prime  vertical. 

Values  of  log  Ra  for  different  latitudes  and  azimuths  will  be 
found  in  Table  XI. 

98.  The  Mean  Value  of  Ra. 

The  mean  value  of  Ra  at  any  point  for  all  azimuths  from  o°  to 
360°  may  be  found  as  follows:  if  the  angular  space  about  any 
point  M  be  divided  into  a  large  number  of  small  parts,  each  equal 
to  da  and  each  expressed  as  a  fraction  of  a  radian,  then  the  num- 

ber of  such  parts  in  a  radian  will  be  —  ,  and  the  number  in  a  cir- 

da 

cumference  will  be  ^~  .    If  the  value  of  Ra  be  computed  for  each 
da 

of  these  azimuths,  then  the  sum  of  these  values  of  Ra,  divided 
by  their  number,  is  the  mean  value;  that  is, 


mean  Ra  =    I      Ra 

JQ 


2TT 

NR 


2Tr  Jo     N  cos2  a  +  Rm  sin2  a 
tffc. 


TT  JQ 


IT  Jo   N  cos2  a  +  Rm  sin2  a 


da. 


132  PROPERTIES  OF  THE  SPHEROID 

To  integrate  this  quantity,  substitute  a  new  variable,  /  = 

tanaV/-?,  from  which  dt  =  l/T?- — T~-    Bv  dividing  both 
*  N  '  N    cos2  a 

numerator  and  denominator  by  N  cos2  a  and  factoring  NRm,  the 

integral  may  be  put  in  the  form 

I — 

T  *      Rm  I  j 

P          2     /-^r,    ^"LZ^i^Z 

mean  #a  =  -  v^m]V  I  — — — , 

.   N  cos2  a 
which,  by  substitution,  becomes 


mean  Ra  =  -  V^mA^  f   - 

7T  Jo      I 


*_ 

+  /2 


=  -  VRmN  [tan-1 

7T 


[48] 

The  mean  radius  of  curvature  is,  therefore,  the  geometric  mean 
of  the  radii  of  curvature  of  the  two  principal  sections. 

99.   Geometric  Proofs. 

Geometric  proofs  of  the  last  two  formulae  will  be  found  in- 
structive. To  find  Ra  geometrically,  imagine  a  tangent  plane 
at  the  point  M  and  also  a  parallel  plane  at  an  infinitesimal  dis- 
tance below  M.  This  second  plane  will  cut  from  the  surface  a 
small  ellipse.  It  has  already  been  shown  that  the  radius  of 
curvature  of  the  prime  vertical  section  is  ^V.  In  Fig.  60  the 
points  A,  M,  and  B  are  on  the  circle  whose  radius  is  N  and  whose 
center  is  the  point  H  on  the  axis.  By  similar  triangles, 

M C  :  CA  =  CA  :  CK. 
Since  MC  is  infinitesimal, 


GEOMETRIC   PROOFS 
similarly,  for  a  section  in  the  meridian 

ur        b* 
=  — — ; 

2  -K-m 

and,  in  general,  for  any  section, 

MC  =-^-- 


133 


The  coordinates  of  the  point  P  (Fig.  61)  are 

x  =  s  •  siri  a        and        y  =  s  •  cos  a. 


FIG.  61. 


Substituting  these  in  the  general  equation  of  the  ellipse, 
s2  sin2  a      s2  cos2  a 

~^~      ~v~     '• 

But,  from  the  preceding  equations, 


-2  =  §         and 

a2      N 


R 


hence 


or 


NRm 


N  cos2  a  +  Rm  sin2  a 


[47] 


PROPERTIES  OF  THE   SPHEROID 


To  show  geometrically  that  the  mean  value  of  Ra  = 
observe  that,  as  before, 


mean  Ra 


-T 

2  IT  JQ 


Ra-da 


and,  from  the  preceding  paragraph, 


Therefore 

But 

Therefore 


mean  Kn  = 


2  IT  JQ 


b2 


da. 


i  r2* 

•I      s2  da  =  area  of  ellipse  =  irab. 

2  JQ 

n 

mean  Ra  =  -  X  irab  X -^ 

TT  0* 


Therefore 


aR, 
b 


mean  Ra  =  VNRm. 


[48] 


100.  Length  of  an  Arc  of  the  Meridian. 

Any  small  arc  of  the  meridian  ellipse  may  be  regarded  as  an  arc 
of  a  circle  whose  radius  is  Rm,  the  error  being  very  small  for  short 
arcs.  The  length,  therefore,  is 


or,  if  d<j)  is  in  seconds  of  arc, 

s  =  Rmd<j>"  •  arc  i".  [49] 

If  the  arc  is  so  long  that  the  value  of  Rm  varies  appreciably,  it  is 
necessary  to  find  5  by  integrating  the  expression 

a  (i  —  e2) 
ds  = * '—: •  dd>     • 


between  the  limits  <£i  and  02. 


LENGTH  OF  ARC  OF  MERIDIAN  135 

If  we  expand  the  denominator  by  the  binomial  theorem,  we 
have 


n60  .  .  .  )  d<t>. 
Integrating, 

•  •  •  )  d<f>. 


In  order  to  integrate  the  terms  of  the  series  in  parenthesis  we 
simplify  the  expression  by  means  of  the  following  relations: 

sin2  0  =  \  —  \  cos  2  0, 

sin4  0  =  f  —  \  cos  2  0  +  \  cos  4  0, 

sin6  0  =  ye  —  if  cos  2  0  +  y  g  cos  4  0  —  -^  cos  6  0. 

Integrating  and  substituting  the  limits,  0i  and  02,  we  have 
5  =  a  (i  —  e2)  { A  (02  —  0i)  —  \  B  (sin  2  02  —  sin  2  0i) 

+ 1  C  (sin  4  02  -  sin  4  0i)  .  .  .    } ,     [50] 

in  which  A  =  1.0051093,  B  =  0.0051202,     and  C  =  0.0000108. 

(See  Jordan's  Handbuch  der  Vermessungskunde,  Vol.  Ill,  p..  226; 
and  Crandall's  Geodesy  and  Least  Squares,  p.  163.) 

101.   Miscellaneous  Formulas. 

The  following  formulas,  relating  to  the  ellipse,  are  given  here 
for  convenience  of  reference. 

The  geocentric  latitude  may  be  found  from  the  expression 

tan  \l/  =  *•  =  (i  —  e2)  tan  0  =  —  tan  0.  [51] 

x  a* 

The  maximum  difference  between  0  and  ^  is  about  o°  n'  40",  at 
latitude  45°.  At  the  equator  and  at  the  poles  the  difference  is 
zero. 

The  reduced  latitude,  0  (see  Art.  94,  p.  123),  may  be  found 
from  the  geodetic  latitude  by  means  of  the  relation 

a  tan  0  =  b  tan  0  [52] 

which  is  readily  proved  from  Fig.  57. 


PROPERTIES   OF   THE    SPHEROID 


The  compression  of  the  spheroid,  that  is,  the  flattening  at  the 
poles,  is  expressed  by 

f-  (sal 


The  length  of  a  quadrant  of  the  meridian  is  given  by 


[54] 


FIG.  62. 

102.  Effect  of  Height  of  Station  on  Azimuth  of  Line. 

Since  the  normals  drawn  from  two  points  on  the  surface  do  not 
in  general  lie  in  the  same  plane,  there  will  be  an  error  in  the 
observed  horizontal  direction  of  a  station,  depending  upon  its 
height  above  the-  surface  of  the  spheriod.  This  error  may  be 
likened  to  the  error  of  sighting  on  an  inclined  range-pole;  the 

*  From  the  equation  for  the  length  of  a  meridian  arc,  we  have  for  the  quadrant 

•JT 

q  =  a  (i  -  c2)  f    (  i  +  ^  t2  (i  -  cos  2  </>)  +  ^  e4  (3  -  4  cos  2  0  +  cos  4  </>)  )  <fy 
•/o    \         4  04  / 


-f     g2  +       ,, 
4  64 


32 


_ 
256 


EFFECT  OF  HEIGHT  OF  STATION  ON  AZIMUTH  OF  LINE     137 

higher  up  the  sight  is  taken,  the  greater  the  error  in  the  horizontal 
angle.  In  Fig.  62*he  observer  is  at  A  and  sighting  at  point  M, 
which  is  at  an  elevation  h  above  sea-level.  The  vertical  plane  of 
the  instrument  projects  M  down  to  sea-level  at  B  on  the  line  MH, 
H  being  the  end  of  the  normal  at  A.  The  point  which  is  verti- 


FIG.  62a.    A  vertical  in  latitude  o°  and  a  vertical  in  latitude  60°; 
d\  =  80°;  e  =  0.81;  (looking  SW). 

cally  below  M  is  B',  as  determined  by  the  normal  ME'.  Denote 
by  6  the  angle  EME'  or,  what  is  nearly  the  same,  HBHf.  The 
angle  (x)  subtended  by  BBf  at  point  A  (the  observer's  position) 
is  the  correction  desired.  The  latitude  of  A  is  0,  and  that  of  M 
is  </>'.  In  the  triangle  MEE' 

sin  5  EE'  EE' 


sin  EE'M      EB  +  BM      EB 


,  . 

(approx.), 


or 


EB 

where  0'  is  the  latitude  of  B'. 


138  PROPERTIES   OF   THE   SPHEROID 

Now       EH'  =  OH'  -  OH 

=  (Nr  -  nf)  sin  0'  -  (N  -  n)  sin  0  j^V 

=  N'#  sin  0'  -  Ne2  sin  0.         Wf5 
^ 

Therefore         5  =          -  (N'e2  sin  0'  -  #e2  sin  0) 


('N'  .  \ 

—  sin  <£'  —  sin  0)> 

AT 
in  which  —  may  be  put  =  i  with  small  error. 

Then  6  =  e2  cos  </>'  (2  cos  -  (0  +  <t>')  sin  —  V 

where          A</>  =  <£'  —  </>; 

Then  6  =  ^cos2^'  X  A<£  (approx.). 


=  e2  cos2  0'  — -  cos  a 

/  9      •      9        /\  8 

2       2  ,f  (i  —  e2  sin2  0  )2 

=  eL  cos2  0  •  ^  cos  a  •  •* ; ~LZ- 

*(i  -  e2) 

The  factor  g   >m          differs  but   little  from  unity 

(i  —  e ) 

may  be  considered  equal  to  unity  in  this  equation. 

™,  e2  •  5  •  cos2  0'  •  cos  a  (  . 

Then  5  =  -  —  (a) 

The  linear  distance  BB'  =  hd,  and  the  correction  to  the  azi- 
muth (x)  3.t  point  A  is  given  by 

hd  sin  a 
s  arc  i" 

/?e2  cos2  0X  sin  o;  cos  a 
a  arc  i" 


arc  i" 


•  e2  •  -  •  sin  2  a  •  cos2  0r,  [55] 


as  given  by  Clarke  (Geodesy,  p.  112).     This  may  be  written 

x"  =  k  •  h  •  sin  2  a  cos2  0'  [56] 


REFRACTION  139 

*2 
where 


2 

\ 


the  dimensions  being  in  meters. 

The  logarithm  of  k  is  6.03920. 

When  the  signal  is  NE  or  SW  of  the  observer  the  azimuth  must 
be  increased  to  obtain  the  correct  azimuth  at  sea-level;  if  the 
signal  is  NW  or  SE  the  observed  azimuth  must  be  decreased. 

If,  when  deriving  the  above  equation,  we  place  the  fraction 

n       •       O      /  / 

— p£  =  i,  the  formula  for  x"  should  have  a  replaced  by  N. 

I  €r 

For  0  =  45°,  a  =  45°,  and  h  =  1000  meters,  the  value  of  x" 
is  o".o547.  This  is  much  smaller  than  the  probable  error  of  an 
observed  direction  (see  p.  65),  and  is  therefore  negligible  except 
for  great  heights.  This  correction  has  been  applied  to  angles 
measured  in  the  primary  triangulation  of  the  California  and 
Texas  arc  and  the  California  and  Washington  arc.  It  is  too 
small  to  affect  the  triangulation  of  the  eastern  half  of  this  country. 

Questions.  —  What  influence  does  the  height  of  the  observer  have  upon  the 
result? 

Why  does  the  distance  not  enter  into  the  formula? 

Which  one  of  the  two  approximations  is  more  accurate,  that  giving  a  in  the 
denominator,  or  that  giving  .AT  ? 

103.  Refraction. 

Inasmuch  as  the  refraction  acts  in  the  vertical  plane  at  any 
point,  and  the  vertical  plane  changes  its  direction  as  the  ray  pro- 
ceeds along  the  line,  it  is  evident  that  there  must  be  some  hori- 
zontal displacement  of  the  object  sighted,  due  to  the  refraction. 
Investigations  show  that  this  error  is  quite  inappreciable  for  all 
lines  that  can  actually  be  observed. 

104.  Curves  on  the  Spheroid.    The  Plane  Curves. 

When  a  theodolite  is  set  up  at  any  point  A  and  leveled,  its 
vertical  axis  is  made  to  coincide  with  the  direction  of  the  normal 
at  A ,  which,  except  for  local  deflections,  coincides  with  the  direc- 
tion of  the  force  of  gravity  at  A.  If  another  theodolite  is  set  up 
at  B,  in  a  different  latitude  and  a  different  longitude,  it  is  evident 


140  PROPERTIES  OF  THE   SPHEROID 

that  these  vertical  axes  are  not  in  the  same  plane,  since  their 
normals  (plumb  lines)  never  intersect.  The  greater  the  latitude, 
the  lower  the  point  where  the  normal  intersects  the  polar  axis. 
It  is  clear  that  the  line  marked  out  on  the  surface  of  the  spheroid 
by  the  line  (or,  rather,  plane)  of  sight  of  the  first  theodolite  is  not 
the  same  as  the  line  marked  out  by  the  vertical  plane  of  the  other 
theodolite.  If  A  is  southwest  of  B,  then  the  curve  cut  by  the 
plane  of  the  theodolite  at  A  is  south  of  that  cut  by  the  plane  of 
sight  of  the  theodolite  at  B.  This  may  be  seen  from  the  fact 
that  both  planes  contain  the  chord  AB;  and  since  the  normal  at 
A  is  higher  at  the  polar  axis,  the  curve  itself  must  be  lower 
(farther  south) . 

105.  The  Geodetic  Line. 

Another  curve  which  holds  an  important  place  in  the  theory 
of  geodesy  is  known  as  the  geodetic  line.  This  is  the  shortest 
line  that  can  be  drawn  on  the  surface  of  the  spheroid  between 
two  given  points.  It  is  not  a  plane  curve,  but  has  a  double  cur- 
vature, A  characteristic  property  of  the  curve  is  that  the  oscu- 
lating plane  *  at  any  point  on  the  curve  contains  the  normal  to 
the  surface  at  that  point.  In  most  cases  the  geodetic  line  is 
found  to  lie  between  the  two  plane  curves  and  has  a  reversed 
curvature.  Fig.  63  is  a  photograph  of  a  model,  the  semi-axes  of 
which  are  a  =  6  inches  and  b  =  3.5  inches.  The  two  plane 
curves  are  shown  and  between  them,  with  the  curvature  slightly 
exaggerated,  is  the  geodetic  line. 

In  order  to  obtain  a  clear  conception  of  the  nature  of  the  ge- 
odetic line,  let  us  imagine  that  a  transit  instrument  is  set  at  point 
A  (Fig.  64),  leveled,  and  then  sighted  at  point  B.  Then  it  is 
moved  to  point  B,  set  up,  and  leveled  again,  and  a  back  sight  is 
taken  on  A]  point  C  is  then  fixed  by  reversing  the  telescope. 
When  the  sight  is  taken  to  A,  the  sight  line  traces  out  the  plane 
curve  BbA ;  and  when  point  C  is  sighted,  it  traces  out  BbC.  The 

*  The  osculating  plane  may  be  considered  to  pass  through  three  consecutive 
points  of  the  curve.  In  reality  it  is  the  limiting  position  approached  by  the  plane 
as  the  distance  between  the  three  points  decreases  indefinitely. 


THE   GEODETIC  LINE 


141 


FIG.  63.    Plane  Curves  and  Geodetic  Line. 


FIG.  64. 


142  PROPERTIES  OF  THE   SPHEROID 

instrument  is  then  taken  to  C  and  the  process  repeated.  It 
should  be  observed  that  the  (vertical)  sight  plane  of  the  instru- 
ment coincides  with  the  normal  to  the  surface  at  each  station. 
If  the  points  A,  B,  C,  D  are  imagined  to  approach  nearer  and 
nearer,  so  that  AB,  BC,  etc.,  become  infinitesimal  elements  of  the 
curve,  the  plane  which  contains  three  consecutive  points  of  the 
curve  also  contains  the  normal  to  the  surface.  If  we  imagine 
the  instrument  to  move  along  this  line,  it  is  seen  that  the  vertical 
plane  of  the  instrument  twists  so  that  it  always  contains  the 
normal. 

One  of  the  characteristic  properties  of  the  geodetic  line  is 
shown  by  the  equation 

Rp  sin  a  =  k,  a  constant  [57] 

Rp  being  the  radius  of  the  parallel  and  a  the  azimuth  of  the  ge- 
odetic line  at  any  point.  This  equation  may  be  derived  analyti- 
cally by  the  methods  of  the  calculus  of  variations  (see  Clarke, 
Geodesy,  p.  125)  or  by  geometric  construction  (see  Jordan,  Ver- 
messungskunde,  Vol.  Ill,  p.  395).  From  this  equation  it  will  be 
seen  that  when  a  is  a  maximum  (90°),  sin  a  =  i  and  Rp  =  k. 
The  constant  of  the  equation  is  therefore  the  radius  of  the  parallel 
of  latitude  beyond  which  the  geodetic  line  does  not  pass.  When 
a  is  a  minimum,  Rp  is  a  maximum,  that  is,  Rp  =  a,  the  equatorial 
radius  of  the  spheroid.  This  shows  that  in  general  a  geodetic 
line  cutting  the  equator  at  any  angle  a  may  go  northward  up  to 
some  (limiting)  parallel  of  latitude  0°  (corresponding  to  Rp  =  k)t 
but  will  not  pass  north  of  this  parallel.  In  the  southern  hemi- 
sphere it  will  reach  a  limit  (—  0°)  having  the  same  numerical 
value.  Such  a  geodetic  line,  when  traced  completely  around  the 
spheroid,  will  not  in  general  return  exactly  on  itself,  but  will  pass 
the  initial  point  on  the  equator  in  a  slightly  different  longitude 
and  then  proceed  to  form  another  loop  around  the  spheroid. 

Except  for  a  few  particular  cases  the  geodetic  line  lies  between 
the  two  plane  curves  and  divides  the  angle  between  them  in  the 
ratio  of  about  2  to  i,  as  shown  in  Fig.  65. 


THE  ALIGNMENT  CURVE  143 

If  the  terminal  points  P  and  Q  are  in  nearly  the  same  latitude, 
the  geodetic  line  may  cross  the  plane  curve. 

It  is  important  to  bear  in  mind  that  the  lengths  of  these 
different  curves  on  the  spheroid  differ  by  quantities  that  are 
quite  inappreciable  in  practice.  The  differences  in  length  are  far 
shorter  than  the  distances  by  which  the  curves  are  separated  at 
their  middle  points  (Art.  107),  and  even  these  latter  are  negligible 
in  practice.  Also  the  angle  by  which  the  azimuth  of  the  geodetic 
differs  from  the  azimuth  of  the  plane  section  is  much  smaller  than 
can  be  measured. 


FIG.  65. 

It  should  be  noted  that  the  geodetic  line  itself  cannot  be  sighted 
over  directly,  because  it  is  not  a  plane  curve,  and  that  the  geodetic 
triangle  can  be  obtained  only  by  computation. 

1 06.  The  Alignment  Curve. 

Another  curve  which  may  be  drawn  on  the  surface  is  denned 
in  the  following  manner:  if  the  theodolite  be  supposed  to  move 
from  A  to  B,  keeping  always  in  line  between  the  two  points  (that 
is,  the  azimuths  of  A  and  B  180°  apart),  and  the  instrument  being 
always  leveled,  its  path  will  be  a  curve  which  lies  very  close  to  the 
geodetic  line  and  generally  between  the  two  plane  curves.  This 
is  called  the  alignment  curve. 

It  is  possible  to  define  other  curves  *  between  these  two  points. 
*  See  Coast  Survey  Report  for  1900,  p.  369. 


144 


PROPERTIES  OF  THE   SPHEROID 


Such  curves  are  of  theoretical  value  only,  since  the  lengths  of  all 
such  lines  on  the  earth's  surface  differ  from  each  other  by  quanti- 
ties too  small  to  measure.  The  two-plane  curves,  however,  are 
separated  by  a  distance  which  is  quite  appreciable. 

107.  Distance  between  Plane  Curves. 

The  maximum  separation  of  the  two  plane  curves  may  be 
computed  approximately  as  follows:  the  angle  (d')  between  the 


FIG.  66. 


two  planes  is  very  nearly  equal  to  the  angle  d  multiplied  by  sin  a, 
since  d  is  the  angle  measured  in  the  plane  of  the  meridian,  whereas 
the  angle  desired  (5',  Fig.  66)  is  that  perpendicular  to  the  planes 
of  sight. 


Therefore 


5'  = 


se2  cos2  </>  cos  a  sin  a 

N 


(see  equation  (a),  p.  138). 

The  distance  of  the  chord  AB  (Fig.  67)  below  the  surface  (D) 
at  its  middle  point  is  given  by 


2         2 


or,  approximately, 


D 


DISTANCE  BETWEEN  PLANE  CURVES 

The  curves  are  separated  at  their  middle  points  by  the  hori- 
zontal distance 

n*'        s2    ^  s#  cos2  <f>  cos  a  sin  a 


— -  e2  cos2  0  cos  a  sin  a. 


[58] 


D 


FIG.  67. 

The  difference  in  azimuth  may  be  computed  approximately 
by  finding  the  angle  between  the  two  tangents  to  the  curve  drawn 
from  one  of  the  stations  and  prolonged  half  the  distance  (Fig.  68). 
The  terminal  points  of  these  tangents  will  be  at  a  distance  D 
above  the  surface  and  will  be  separated  by  a  distance  2  D8'.  The 
angle  between  these  two  lines  is  nearly 

2P8' 
k  s  arc  i" 


2  5s    e2  cos2  #  cos  a  sin  a 

isarci" 
e2  cos2     cos  a  sin  « 


arc  i 


[59] 


146  PROPERTIES  OF  THE  SPHEROID 

For  the  oblique  boundary  line  between  California  and  Nevada  * 
5  =  650,000  m.,  (400  mi.),  <f>m  =  37°  oo',  a  =  134°  33';  whence 
Dd'  =  1.8  meters  and  the  difference  in  azimuth  =  2^. . 


2D8' 


FIG.  68. 

For  the  western  boundary  of  Massachusetts  s  =  80,930  m., 
(50  mi.),  <f>m  =  42°  24',  a  =  195°  12';  this  gives  Dd'  =  0.0015 
meter  and  Aa  =  o".oi6. 

PROBLEMS 

Problem  i.  Prove  by  the  process  outlined  in  the  first  paragraph  of  Art.  96  that 
the  radius  of  curvature  of  the  prime  vertical  section  of  the  spheroid  is  N,  the  nor- 
mal terminating  in  the  minor  axis. 

Problem  2.  A  model  of  the  spheroid  has  an  equatorial  diameter  of  12  ins.  and  a 
polar  diameter  of  7  ins.  Compute  the  correction  to  reduce  to  "sea-level"  the 
azimuth  of  a  line  in  latitude  45°,  the  azimuth  being  45°  and  the  elevation  of  object 
being  one  inch  above  the  surface  of  the  spheroid. 

Problem  3.  What  will  be  the  maximum  separation  of  two  plane  curves  drawn 
on  the  model  described  in  problem  2  if  s  =  7.5  ins.,  mean  <£  =  30°,  a  =  45°? 
(Use  the  approximate  formula.) 

*  See  Coast  Survey  Report  for  1900,  p.  368. 


CHAPTER  VI 
CALCULATION  OF  TRIANGULATION 

108.  Preparation  of  the  Data. 

From  the  records  of  the  field-work  of  the  triangulation  we  ob- 
tain a  value  for  each  angle,  supposed  to  be  freed  from  the  errors 
of  the  instrument,  eccentricity  of  station,  phase  of  signal,  eleva- 
tion of  signal,  etc.  Before  these  angles  are  employed  for  solving 
the  triangles,  they  should  be  examined  to  see  if  they  satisfy  any 
geometric  conditions  existing  among  them.  If  at  any  station 
two  or  more  angles  and  their  sum  have  been  measured,  then  these 
angles  must  be  so  corrected  that  they  exactly  equal  their  sum. 
If  the  horizon  has  been  closed,  the  measured  angles  must  be  ad- 
justed so  that  their  sum  equals  360°.  If  the  angles  have  been 
measured  with  different  degrees  of  precision,  as,  for  example, 
with  different  instruments  or  a  different  number  of  sets  or  of 
repetitions,  the  different  angles  should  be  given  proper  weights; 
and  if  the  best  possible  values  are  desired,  the  angles  at  each 
station  should  be  adjusted  by  the  method  of  least  squares. 

After  the  station  adjustment,  as  it  is  called,  has  been  completed, 
the  triangles  must  be  examined  to  see  if  the  sum  of  the  three 
angles  in  each  triangle  fulfills  the  requirement  that  this  sum 
shall  equal  180°  plus  the  spherical  excess  of  the  triangle.  The 
verticals  at  the  three  triangulation  stations  are  not  parallel  to 
each  other,  because  the  surface  is  curved.  Consequently  the 
sum  of  the  angles  will  exceed  180°  by  an  amount  which,  on  a 
spherical  surface,  would  be  exactly  proportional,  and  which,  on  a 
spheroidal  surface,  is  nearly  proportional  to  the  area  of  the  tri- 
angle. 

As  was  shown  in  the  preceding  chapter  (Art.  102),  the  error  in  the 
direction  of  an  object,  due  to  the  fact  that  the  earth  is  spheroidal 

147 


148 


CALCULATION  OF  TRIANGULATION 


instead  of  spherical,  is  extremely  small,  even  when  the  object 
is  several  thousand  meters  above  sea-level.  Hence  it  follows 
that  if  the  vertices  of  a  spheroidal  triangle  are  projected  vertically 
onto  the  surface  of  a  tangent  sphere,*  the  errors  thus  produced 
in  the  horizontal  angles  of  the  triangle  will  be  much  less  than  the 
errors  in  the  measurement  of  the  angles,  because  the  points  on 
the  sphere  and  those  on  the  spheroid  are  separated  by  compara- 
tively short  distances.  This  enables  us  to  compute  spheroidal 
triangles  as  spherical  triangles  and  greatly  simplifies  the  com- 
putation. The  lengths  of  the  triangle  sides  will  be  practically 
the  same  on  the  two  surfaces. 

In  this  connection  it  is  well  to  bear  in  mind  that  if  the  topog- 
raphy of  the  earth's  surface  were  represented  on  an  1 8-inch 
globe  the  total  variation  in  elevation  would  scarcely  be  greater 
than  the  thickness  of  a  coat  of  varnish.  The  elevation  of  the 
geoid  above  the  spheroid  would  be  very  much  smaller  than  this, 

and  the  distance  between  the 
spheroid  and  the  tang^^ 
sphere  at  any  station  would 
usually  be  still  smaller.  This 
will  give  some  idea  of  the 
minuteness  of  the  errors  under 
discussion. 

It  should  be  remembered 
that,  whereas  the  triangulation 
stations  themselves  are  at  va- 
rious heights  above  sea-level, 

these  are  all  supposed  to  have  been  projected  down  vertkally  onto 
the  spheroid  before  beginning  the  computation  of  the  triangle. 
The  points  of  which  we  shall  speak  in  discussing  the  solution  of 
the  triangles  and  the  geographical  positions  of  the  stations  are 
these  points  on  the  spheroidal  surface  and  not  the  original 
station  points. 

*  The  sphere  is  supposed  to  be  tangent  at  the  center  of  gravity  of  the  triangle  to 
be  computed. 


FIG.  68a. 


SPHERICAL  EXCESS  149 

In  solving  triangles  by  the  methods  given  below,  the  following 
approximations  have  been  made,  and  it  is  assumed  that  in  all 
cases  the  resulting  errors  are  negligible. 

i.  The  reduction  to  sea-level  reduces  the  observed  direction 
to  that  corresponding  to  the  geoid  (or  actual  surface),  not  the 
spheroid,  as  is  assumed. 

'2.  The  effect  of  local  deflection  of  the  plumb  line  is  not  allowed 
for. 

3.  The  effect  of  atmospheric  refraction  on  the  direction  (hori- 
zontal refraction)  is  neglected. 

4.  The  reduction  of  the  observed  direction  (plane  curve)  to 
that  of  the  geodetic,  or  shortest,  line  is  omitted.     There  are  in 
reality  eight  triangles  formed  by  the  plane  curves,  which  are 
treated  as  if  they  were  identical  (see  Art.  104). 

109.  Solution  of  a  Spherical  Triangle  by  Means  of  an  Auxiliary 
Plane  Triangle. 

The  direct  solution  of  the  triangles  of  a  net  as  spherical  triangles 
j^uld  be  unnecessarily  complicated.  This  may  be  avoided  by 
employing  a  principle  known  as  Legendre's  Theorem,  namely, 
that  if  we  have  a  spherical  triangle  whose  sides  are  short  com- 
pared with  the  radius  of  the  sphere,  and  also  a  plane  triangle 
whose  sides  are  equal  in  length  to  the  corresponding  sides  of  the 
spherical  triangle,  then  the  corresponding  angles  of  the  two  tri- 
angles differ  by  approximately  the  same  quantity,  which  is  one- 
third  of  the  spherical  excess  of  the  triangle. 

no.   Spherical  Excess. 

The  spherical  excess  of  a  triangle  is  directly  proportional  to  its 
area,  as  shown  in  spherical  geometry.  Hence,  if  A'  is  the  area 
of  any  triangle,  R  is  the  radius  of  the  sphere,  S  is  the  surface  of 
the  sphere,  and  e  is  the  spherical  excess  of  the  triangle;  then, 

since  the  spherical  excess  of  the  tri-rectangular  triangle  is  - , 


150  CALCULATION  OF  TRIANGULATION 

ie      2  Af 

7  ;= 

A' 

Therefore  e  =  —  • 

To  express  e  in  seconds  of  arc,  divide  by  arc  i",  and  we  have 

„  A'  be  sin  A  r,  -, 

•6    =F^7>  =  2^arci"' 

where  b,  c,  and  A  are  two  sides  and  the  included  angle  of  the  tri- 
angle, a  and  b  being  in  linear  units. 

The  sphere  which  is  tangent  to  the  spheroid  at  the  center  of 
gravity  of  the  triangle,  and  which  has  the  same  average  curva- 
ture, is  a  sphere  of  radius  =  VRmN',  whence 

„          be  sin  A  ,     .  r,  , 

e    =     T)  *r  -  7-.  =  mbcsmA.  [6i\ 

2RmNsuci" 

The  quantity  -—  —  -  =  m  is  given  for  different  latitudes 

2  RmN  arc  i 

in  Table  XII.     The  latitude  to  be  used  in  finding  m  is  the  mean 
of  the  latitudes  of  the  three  vertices  of  the  triangle. 

Questions,  —  Is  this  auxiliary  plane  triangle  the  same  as  the  chord  triangle 
formed  by  joining  the  points  by  straight  lines?  Are  the  two  similar  in  shape? 

in.  Proof  of  Legendre's  Theorem. 

To  prove  Legendre's  theorem,  let  A',  B'  and  C  be  the  angles 
of  the  spherical  triangle,  and  A,  B,  and  C  those  of  the  plane  tri- 
angle; the  sides  of  the  plane  triangle  are  a,  b,  and  c,  and  those  of 
the  spherical  triangle  are  a'R,  b'R,  and  c'R,  then,  in  the  plane 
triangle, 

tf  +  c2  -  a2 
cos  A  =  --  -  --  >  (a) 


or 


-  a*  -  b*  -  c* 


PROOF  OF  LEGENDRE'S  THEOREM  151 

In  the  spherical  triangle, 

A  f      cos  a'  —  cos  6'  cos  c' 

cos  A    =  ; — 1-7—; — -, 

sin  6  sm  c 

Expanding  each  sine  and  cosine  (omitting  terms  of  higher 
order  than  the  fourth), 

a'2      a'4      /        b'*^b'*\(       C'2-L-C'*\ 
i 1 i 1 lli 1 1 

. ,  2  24         \  2  24/  \  2  24/ 


k  (-  a'2  +  b"1  +  c'1)  -  A  (6'*  +  C'4  -  o'4)  -  j  6V2 


=  ft  (  -  a'2  +  ft'2  +  c'2)  -  A  (&'"  +  c'4  -  a'4)  -  i  6V2]  I+, 

t?  C 

'c/4  _  fl/4  +  6  &  y2 


26V  24  6V 

-  a'W  +  6/4  +  2  6V2  -  a;  V2  + 


1  2  6V 


whence    cos  .4'  = 

+  2  arV/2  +  2 


2  6V'  6  4  6V 

From  (a),  (6),  and  (c) 

cos  ^4'  =  cos  A  —  \  6V  sin2  A. 
Let  #  be  the  difference  between  A  and  A1  '.     Then 
cos  x  =  i  and  sin  x  =  x"  arc  i"  (nearly),  since  x  is  small, 

and  cos  A'  =  cos  (A  +  #) 

=  cos  A  —  sin  ^Iz"  arc  i/; 
=  cos  ^4  —  i  6V  sin2  A  ; 

that  is,        x"  arc  i"  sin  ,4  =  \  6V  sin2  ^. 

r™       r  //      6Vsinyl 

Therefore  x"  =  -     —  77  > 

6  arc  i 


IJ2  CALCULATION  OF  TRIANGULATION 

or,  since  bf  =  -     and    c'  =  •£> 

/v  /v. 

„         be  sin  ^4  r  £  , 

=  6  IP  are  i" '  [62] 

It  will  be  noticed  that  this  is  one-third  of  the  spherical  excess 
as  found  in  Equa.  [60].  The  same  result  would  also  be  found 
for  angles  B  and  C. 

112.  Error  of  Legendre's  Theorem. 

The  error  in  Legendre's  theorem  *  as  applied  to  the  sphere  may 
be  studied  by  carrying  out  the  above  series  so  as  to  include  terms 
of  higher  powers  than  the  fourth.  Jordan  (Vermessungskunde) 
gives  a  numerical  example  showing  the  amount  of  this  error  in  a 
triangle  of  which  the  side  AC  is  about  65  miles  in  length;  the 
angles  are  shown  below: 

,4 '  =  40°  39'-3<>".38o 
B'  =  86  13  58  .840 
C'  =  53  06  45  .630 

180°  oo'  i4".85o 

Denoting  the  spherical  angles  by  A',  B'}  Cf,  and  the  correspond- 
ing plane  angles  by  A,  B}  C,  the  differences  are  as  follows,  the  first 
column  containing  the  values  derived  from  Legendre's  theorem 
in  its  ordinary  form,  the  second  containing  the  smaller  terms 
which  are  usually  neglected. 

Approx.  Exact. 

A'  —  A  4".Q5ooi8  4.950036 

B'  —  B  4  ..950018  4.949997 

C'  —  C  4  .950018  4.950021 

113.  Calculation  of  Spheroidal  Triangles  as  Spherical  Tri- 
angles. 

It  is  customary  to  assume  that  the  differences  between  the 
spherical  and  spheroidal  triangles  are  negligible  when  the  actual 
points  are  projected  down  onto  a  tangent  sphere  of  radius  VRmN. 
Clarke,  in  his  Geodesy,  shows  the  error  of  this  assumption  in  the 
case  of  a  triangle  having  a  side  over  200  miles  long,  the  result 
being  as  follows: 

*  See  Coast  Survey  Special  Publication  No.  4,  p.  51. 


CALCULATION  OF  THE   PLANE  TRIANGLE 


153 


A' 
B' 
C' 

e' 

Spheroidal 
98°  44'  37".096| 
58°  16'  46".S994 

23°   OO'    I2".7303 

A 
B 
C 

e 

Spherical 

98°  44'  37"-i899 
58°  16'  46".4737 
23°  oo'  I2".7634 

The  preceding  example  indicates  that  in  triangles  composed  of 
lines  such  as  can  be  sighted  over  on  the  earth's  surface  the  error 
involved  in  computing  spheroidal  triangles  as  spherical  triangles 
is  negligible  in  practice. 

114.  Calculation  of  the  Plane  Triangle. 

After  the  spherical  excess  has  been  computed,  the  angles  of  an 
auxiliary  plane  triangle  may  be  found  by  applying  Legendre's 
theorem,  that  is,  by  deducting  one- third  of  the  spherical  excess 
from  each  spherical  angle.  The  difference  between  the  sum  of 
these  plane  angles  and  180°  is  the  error  of  measurement  and  may 
be  distributed  equally  among  the  three  angles  unless  a  least- 
square  adjustment  is  to  be  made.  In  any  case  this  method  of 
distributing  the  error  may  be  used  for  a  preliminary  determina- 
tion of  the  distances.  The  lengths  of  the  triangle  sides  are  now 
found  by  plane  trigonometry.  Since  all  three  angles  of  a  tri- 
angle will  usually  be  known,  the  only  formula  that  will  be  used, 
except  in  rare  cases,  is  the  sine  formula, 

a  _  sin  A 
b  ~  sinB 

A  convenient  arrangement  of  this  computation,  used  by  the 
Coast  and  Geodetic  Survey,  is  shown  in  the  following  table.  The 
spherical  excess  of  the  triangle  in  this  case  is  o".86,  which  give? 
i ".2  as  the  error  of  closure  of  the  triangle. 


Stations. 

Observed 
angles. 

Correc- 
tion. 

Spheri- 
cal 
angles. 

Spheri- 
cal 
excess. 

Plane 
angles  and 
distances. 

Loga- 
rithms. 

Blue  Hill  to  Prospect  

o      /      n 

61  47  18  8 

it 

n 

18  4 

n 

22723.  08  m. 

0         t            II 

4.356  4673 

Blue  Hill  

35  45  15.4 

0.4 

15  o 

o  3 

35  45  *4  7 

9  766  6415 

Prospect  

82  27  27.9 

0.4 

27.5 

0.3 

82  27  27.2 

9.9962261 

Observatory  to  Prospect  .  .  . 
Observatory  to  Blue  Hill.  . 

180  oo  02.1 

15067.13 
25563.20 

4.178  0306 
4.4076152 

154 


CALCULATION  OF  TRIANGULATION 


US.  Second  Method  of  Solution  by  Means  of  an  Auxiliary 
Plane  Triangle.* 

Another  method  of  solution  which  has  been  used  to  some  ex- 
tent in  Europe  is  as  follows: 

Let  ABC  (Fig.  69)  be  the  spherical  triangle  and  A'B'X  an 
auxiliary  plane  triangle  having  two  of  its  angles,  a  and  j3,  equal  to 
the  corresponding  angles  in  the  spherical  triangle.  Evidently 
the  third  angles  will  not  be  equal. 

A' 


FIG.  69. 

Let  a'  and  b'  in  the  plane  triangle  be  the  sides  corresponding 
to  a  and  b. 

In  the  spherical  triangle  we  have 

.    a 

sm- 

sm  a  R 


sin/3        .    b' 
sm- 


and  in  the  plane  triangle 


sin  a      a 


sin  |3      b' 
for  all  values  that  may  be  given  to  a'  and  b'\   whence 


.    a  a 

sin—        , 

—  =  °L  =  ^ 
b  ~  b'~  b'' 


sm- 


R 


*  See  Jordan,  Vermessungskunde,  Vol.  Ill,  39. 


SECOND  METHOD  OF  SOLUTION  155 


This  equation  is  satisfied  if  we  place 

^L  -.    '   i 

V       .    b 
and  —  =  sin  — • 

The  general  expression  for  any  triangle  side  may  be  written 

--sin- 
s' being  the  side  of  an  auxiliary  plane  triangle  corresponding  to 
the  side  s  of  the  spherical  triangle. 

Taking  logs  of  both  members, 

v  ^  i    ^  ^^ 

log-  =  log  sin-  =  log!—  -  —-=.  - 

J\.  K.  Vtv        0  A 


Now,  since 


2       3 

(where  M  =  loge  10  =  0.4342945,  the  modulus  of  the  common 
logarithms),  we  may  write 

,     s'      ,       .    s      ,      s 
log  -  =  log  sin-  =  log- 


\      Mf     ^_Y 

••/  *\  OT  '  •• 


Ms2 


,       .?:       ,      5'       Ms2 
Therefore  log  -  -  log  -  =  — 


/  r,.  i 

or  log  s-  logs   =-2>  163J 


which  is  the  correction  to  the  log  of  the  triangle  side. 

*  The  next  term  «•  -3-  •  51  ••  o.ooo  ooo  oooi  for  a  distance  of  100  kilometers. 
loo    K* 


156 


CALCULATION  OF  TRIANGULATION 


In  calculating  this  correction,  R2  should  be  replaced  by  RmN. 
Values  of  these  corrections  will  be  found  in  Table  XIII  for  the 
argument  log  s. 

Example. 


Stations. 

Spherical  angles. 

Distances. 

Logarithms. 

Blue  Hill  to  Prospect.  .... 

0              1              II 

22,723  .08 

4  -35^  4673 

Correction 

g 

s' 

4   -2c6  4.664 

Observatory 

61  47  18  4 

O  CX4  Q2IS 

Blue  Hill  

•?s  4^  i^  .0 

9  .766  6423 

Prospect  

82  27  27.5 

Q.QQ6  2262 

s' 

4    I?8  O3O2 

Correction 

4 

Observatory  to  Prospect.  . 

15,067.13 

4.178  0306 
4.407  6141 

Correction  

II 

Observatory  to  Blue  Hill  .  . 

25,563.20 

4.407  6152 

Notice  that  after  the  base  of  the  first  triangle  has  once  been 
reduced  by  subtracting  the  correction,  the  computation  of  the 
whole  chain  of  triangles  may  be  carried  out,  using  the  spherical 
angles  only.  It  is  not  necessary  to  add  the  corrections  to  the 
logarithms  of  the  computed  sides  until  their  true  values  are  to  be 
found. 

PROBLEMS 

Problem  i.  Compute  the  area  in  square  miles  of  a  triangle  on  the  earth's  sur- 
face having  a  spherical  excess  of  i",  assuming  that  the  earth  is  a  sphere  of  radius 
3960  miles. 

Problem  2.     Compute  the  sides  of  the  following  triangles: 


Correction  to 

angles  from 

figure  adjustment. 


Error  of 
closure  of 
triangle. 


Corrected 
spherical  angles. 


Spherical 


Station. 

(a)    Mt.  Ellen 
Tushar 
Wasatch 

Wasatch  to  Mt.  Ellen;  azimuth,  333°oi'o8".65;  back-azimuth,  153°  25'o5".oo; 
dist.  123,556.70  meters;  logarithm,  5.0918663.  Latitude  of  Wasatch,  39°  06'- 
54".364;  longitude,  111°  27'  n".9i5. 

(&)    Uncompahgre        -f-o'/.i7 
Mt.  Waas  —  o  .10 

Tavaputs  +o  .58 


-o".7o      )  (    49°  36'  36".88    ) 

+o  .98  +o".22      \    55    56    26  .70 

— o  .06      )  (     74    27    30  .75     ) 


(    31"  54'  6i".S7     ) 

-fo".6s  98    16    41  -16     I       46".i5 

f     49    48    63  .42     ) 


PROBLEMS  157 

Mt.  Waas  to  Uncompahgre;  azimuth,  288°  01'  25".7i;  back-azimuth,  109°  07'- 
o6".n;  dist.  162,928.01  meters,  logarithm,  5.2119958.  Latitude  Mt.  Waas, 
38°  32'  2i".444;  longitude,  109°  13'  38".3O2. 

Problem  3.     Position  of  point  B    j    ^"  39,0  T3,  ^"'ctfi 
Position  of  point  C 


Azimuth  B  to  C  353°  17'  2i".8i;  dist.  40232.35  meters;  (^"=4.6045754); 
back-azimuth  173°  19'  24/'.64. 

The  spherical  angles  are    4  57°  53'  14^.39  (A  is  east  of  BC.) 

B  62°  23'  3i".4o 
C  59°  43'  i7"-93 

Compute  the  spherical  excess  and  solve  the  triangle. 

,  Problem  4.  Position  of  pt.  L\  latitude  42°  26'  i3".276,  longitude  70°  55'52".o88. 
Distance  L  to  N,  3012.0  meters  (log  =  3.478  8600).  Azimuth  L  to  N,  314°  34'  oo"; 
back-azimuth,  134°  35'  03".  Position  of  pt.  N,  latitude  42°  25'  04".764,  longitude 
70°  54'  i8".232.  Angle  at  L,  36°  15' 07";  at  N,  63°  44' 59";  at  E,  79°  59'  5?"- 
(E  is  east  of  LN.)  Compute  the  spherical  excess  and  solve  the  triangle. 

Problem  5.  The  observed  angles  of  a  triangle  and  their  corrections  as  found  by 
adjustment  are  as  follows: 

Angle.  Corrections. 

Sand  Hill  40°  57'  28".i3  -o".3S 

Rutherford  54    22    59  .51  —  o  .61 

Miller  84    39    35  .03  — o  .44 

The  position  of  Rutherford  is  latitude  =  37°  08'  57"-928  N,  longitude  = 
98°  06'  3i".6i8  W.  The  position  of  Miller  is  latitude  =  37°  02'  20^.963  N, 
longitude  97°  55'  43"-9o8  W.  The  azimuth  from  Miller  to  Rutherford  =  127°  28'- 
i7"-95;  back-azimuth  307°  21'  47".3o.  Distance  in  meters,  20139.64;  logarithm, 
4.3040518.  Solve  the  triangle. 

Problem  6.  Show  that  the  substitution  of  .Equa.  (b)  p.  150  in  Equa.  (c)  p.  151 
is  permissible  under  the  assumptions  made  in  Arts.  109  and  in. 


CHAPTER  VII 
CALCULATION  OF  GEODETIC  POSITIONS 

1 1 6.   Calculation  of  Geodetic  Positions. 

In  geodetic  surveys  covering  large  areas  the  positions  of  the 
triangulation  points  are  expressed  by  means  of  their  latitudes 
and  longitudes.  Over  limited  areas .  a  system  of  rectangular 
spherical  coordinates  may  be  used  to  advantage,  but  for  such 
areas  as  have  to  be  surveyed  in  this  country  the  latitude  and 
longitude  system  is  preferable. 

Before  the  latitude  and  longitude  of  one  triangulation  station 
can  be  calculated  from  the  coordinates  of  another  station,  it  is 
necessary  to  know  the  dimensions  of  the  spheroid  which  is  taken 
to  represent  the  earth's  figure,  and  also  to  fix  definitely  the  lati- 
tude and  longitude  of  some  specified  station,  as  well  as  the 
azimuth  of  the  direction  to  some  other  triangulation  station. 
This  selected  position  and  direction  determine  the  relative  posi- 
tion of  the  whole  survey  with  respect  to  the  adopted  spheroid, 
and  constitute  what  is  known  as  the  geodetic  datum.  The  surveys 
of  different  countries  may  be  computed  on  different  spheroids 
or  may  be  located  inconsistently  on  the  same  spheroid.  The 
different  portions  of  a  survey  of  the  same  country  will  be  located 
inconsistently  on  the  same  spheroid  until  they  have  been  con- 
nected by  triangulation. 

The  two  spheroids  which  have  been  most  extensively  used  for 
geodetic  surveys  are  (i)  that  computed  by  Bessel  in  1841,  and  (2) 
that  by  Clarke  in  1866.  The  Bessel  spheroid  was  computed  from 
data  obtained  chiefly  on  the  continent  of  Europe,  and  conse- 
quently conforms  closely  to  the  curvature  of  that  portion  of  the 
earth.  This  spheroid  is  still  in  general  use  in  Europe.  Clarke's 
spheroid  of  1866  was  computed  from  arcs  distributed  over  a  much 

158 


THE- NORTH  AMERICAN  DATUM  159 

larger  portion  of  the  earth's  surface;  it  shows  a  greater  amount 
of  flattening  at  the  poles  than  the  Bessel  spheroid,  and  conse- 
quently assigns  a  flatter  curvature  to  the.  surf  ace  in  the  latitude 
of  Europe  and  of  the  United  States.  The  Bessel  spheroid  was 
employed  by  the  Coast  Survey  in  the  earlier  years.  As  the  sur- 
veys gradually  extended,  the  errors  due  to  using  this  spheroid 
became  more  and  more  apparent,  until  finally,  in  1880,  it  was 
decided  to  change  to  the  Clarke  spheroid.  The  latter  conforms 
much  more  nearly  to  the  curvature  of  the  surface  in  the  United 
States. 

117.  The  North  American  Datum.* 

In  1901  the  United  States  Coast  and  Geodetic  Survey 
adopted  what  was  then  called  the  United  States  Standard  Datum, 
by  assigning  to  the  station  Meade's  Ranch  the  following  position 
on  the  Clarke  spheroid: 

Latitude,  39°  13'  26".686 

Longitude,  98°  32'  3o".5o6 

Azimuth  to  Waldo,    75°  28'  14". 52 

In  1913  this  datum  was  adopted  by  the  governments  of  Canada 
and  Mexico,  and  it  is  now  known  as  the  North  American 
Datum. 

In  deciding  upon  a  geodetic  datum  it  was  necessary  to  con- 
sider two  important  points:  first,  the  datum  should  be  so  chosen 
as  to  reduce  to  a  minimum  the  labor  of  recomputing  the  geodetic 
positions;  second,  it  must  place  the  triangulation  system  in  such 
a  position  that  no  serious  error  will  occur  in  any  part  of  the  sys- 
tem. At  the  time  this  datum  was  selected  there  was  a  large 
number  of  triangulation  points  located  along  the  Atlantic  Coast. 
By  selecting  a  position  for  Meade's  Ranch  consistent  with  the  old 
datum  upon  which  this  triangulation  was  calculated,  a  large 
amount  of  recomputation  was  avoided.  At  the  same  time  it  was 
apparent  that  this  also  placed  the  triangulation  very  near  to  its 
theoretically  best  position. 

*  See  Coast  Survey  Special  Publication  No.  24,  p.  8,  or  Special  Publication  No. 
19,  P-  80. 


i6o 


CALCULATION  OF   GEODETIC  POSITIONS 


118.   Method  of  Computing  Latitude  and  Longitude. 

Assuming  that  the  latitude  and  longitude  of  a  station  (A)  are 
known,  as  well  as  the  distance  and  azimuth  to  a  second  station 
(B),  we  will  now  develop  the  formulae  *  necessary  to  compute 
the  geodetic  latitude  and  longitude  of  the  second  point  In 
doing  this  we  shall  have  to  solve  the  differential  spherical  tri- 
angle formed  by  joining  the  two  points  with  the  pole. 


FIG.  70. 

119.  Difference  in  Latitude. 

In  Fig.  70,  P'  is  the  pole  of  the  spheroid.  P  is  the  pole  of  a 
sphere  tangent  to  the  spheroid  along  the  parallel  of  latitude 
through  A.  The  radius  of  the  sphere  is  N,  and  its  center  is  at 
H.  Let  A  be  the  known  station  and  B  the  unknown  station. 

*  These  formulae  were  first  given  by  Puissant;  see  his  Traite  de  Geodesic,  Vol.  I; 
see  also  Coast  and  Geodetic  Survey  Report  for  1894,  and  Special  Publication  No.  8. 


DIFFERENCE  IN  LATITUDE  l6l 

The  angular  distance  of  A  from  the  pole  is  7;  the  unknown  dis- 
tance of  B  is  7'  ;  a  is  the  arc  AB  ;  a  is  the  azimuth;  and  €  =  180°  —  a. 

If  7'  is  computed  by  a  direct  solution  of  the  spherical  triangle 
ABP,  the  required  precision  can  be  reached  only  by  the  use  of 
about  ten-place  logarithms.  It  is  more  convenient,  and  quite  as 
accurate,  for  such  short  lines  as  occur  in  practice,  to  employ  for- 
mulae giving  the  difference  in  latitude,  that  is  7  —  7'. 

The  formula  for  the  direct  >  solution  of  7'  in  the  spherical  tri- 
angle is 

cos  7'  =  cos  7  cos  a  +  sin  7  sin  a  cos  e.  (a) 

Since  7'  is  a  function  of  0,  its  value  may  be  expressed  as  a  con- 
verging series  by  means  of  Maclaurin's  formula,  giving 


7    =7<r=0        l  --  <      ---  Tl  --  ^^3  --  "   "   "   ' 

a<7ff=o  2    d<j<r=o  o  a<rv=o 

To  evaluate  the  three  differential  coefficients,  differentiate 
Equa.  (a)  three  times  in  succession,  and  in  each  resulting  equation 
substitute  <r  =  o.  The  results  of  the  first  two  differentiations 
are  as  follows: 

—  sin  7'  —  =  —cos  7  sin  o-  +  sin  7  cos  a  cos  e,  (c) 

da 

,d?y'  ,fdv'\* 

—  sin  7  -~v  —  cos  7  I    -     =  —cos  7  cos  o-  —  sin  7  sin  a  cos  e 
da2  \da  / 

=  -cos  7',  (by  (a)).  (<f) 

Before  differentiating  a  third  time,  (d)  may  be  written 

-•£+(£)'-  « 

Differentiating  (e),  we  have 


tan    '         +  sec2    '  ..         +  2        .         =  o 
{/(r3  </<r     da2  da     da2 

When  a  =  o,         7'  =  7, 

and  (c)  becomes 

.      <*7 

—  sin  7  -^  =  sin  7  cos  €. 
da 


1 62  CALCULATION  OF  GEODETIC  POSITIONS 

dy 

Therefore  -~-  =  —  cos  e.  (g) 

da- 

(e)  becomes 

d^y 

tan  7-—  +  cos2e  =  i. 
do2 

Therefore  f*  =  sin2  e  cot  y.  (ti) 

d<r 

(f)  becomes 

x73/y 

tan  7  —  -f  sec2  y  (  —  cos  e)  (sin2  e  cot  7)  +  2  ( —  cos  e)  (sin2  €  cot  7)  =  o. 
dff 

d?y 

Therefore          r~  =  cos  e  sin2  e  cot2  7  (2  +  sec2  7) 
a<r 

=  (2  cot2  7  +  cosec2  7)  sin2  e  cos  c 

=  (i  +  3  cot2  7)  sin2  e  cos  e.  (i) 

Substituting  these  results,  (g),  (ti),  and  (i)  in  equation  (6), 
Maclaurin's  series,  we  obtain 

2  3 

7' =  7  —  0-coseH — •  sin2  e  cot  7  +  — -(i-f  3cot27)sin2ecos€+  •  •  •  .  (j) 
Changing  to  latitudes  and  azimuths  by  placing 

y=  90°  -<*>', 

7  =     90°  -  0, , 
€  =  180°  -  a, 
Equation  (j)  becomes 

0  —  0'  =  a  cos  a  +  ~~  sin2  a  tan  <^> 
2 

~  T  (x +3  tan2  0)  sin2  a  cos  a  ...      (&) 
o 

In  order  to  transfer  the  coordinates  of  the  triangulation  points 
from  the  sphere  to  the  spheroid,  it  should  be  noticed  that  if  the 
radius  of  the  sphere  is  N  (the  normal)  and  its  center  is  at  H  (Fig. 
70),  and  the  polar  axes  of  the  sphere  and  spheroid  coincide,  then 
the  parallels  of  latitude  through  A  coincide,  the  spheroid  being 
tangent  to  the  sphere  along  this  parallel;  also,  the  latitude  (0) 


DIFFERENCE  IN  LATITUDE  163 

will  be  the  same  for  both  surfaces,  and  the  distances  and  azimuths 
of  AB  on  the  two  will  differ  by  inappreciable  quantities.     We 

may  therefore  put  a  =  —  ,  where  s  is  the  distance  in  linear  units. 
Then  (k)  becomes 

0-0'  =         a-f—  ^sin2atan0--sin2«cosc*(i  +3  tan20). 


The  difference  in  latitude  should  be  measured,  however,  on  a 
curve  of  radius  Rm,  since  it  is  measured  along  a  meridian.  The 
linear  difference  in  latitude  is  nearly  the  same  for  the  two  sur- 
faces, and  the  angular  difference  in  latitude  will  vary  inversely  as 
the  radii;  that  is, 

(0  -  0')  #  =  A0"  RM  arc  i".  <m) 

Therefore  ,  A0"  =  (0  -  0')  _    N     „, 

KM  arc  i 

A0"  being  in  seconds  of  arc  on  the  spheroid,  and  RM  the  radius 
of  curvature  of  the  meridian  at  the  middle  point  between  the 
parallels  through  A  and  B.  The  difference  in  latitude  is  there- 
fore 


scosa     ,  S2sin2atan0  _.  .  . 

2  NRMzrci"  ~          6  N2RMzrc  i" 


Since  the  middle  latitude  is  not  known  at  the  beginning  of  the 
computation,  it  is  more  convenient  first  to  take  out  the  value  of 
RTO  for  the  known  latitude  of  A,  giving  60",  and  then  to  correct 
to  RM  by  changing  60"  to  A0"  in  the  inverse  ratio  of  the  radii. 

50"      A0" 
Since  -  =  --, 


M 


in 

in  which  60"  —  —  is  a  correction  to  be  subtracted  from  the  first 

KM 
value. 


164  CALCULATION  OF   GEODETIC  POSITIONS 

From  [43]       Rm  =      ^". 


Therefore        ^  =  a  *  " 


(i  -  62  sin  0)2 

Since  ^1?TO  is  half  the  change  from  the  starting  point  to  the 
middle  point,  d(f>  is  taken  as  half  the  difference  in  latitude,  60; 
that  is, 

,.      50  arc  i" 
a<p  =  —  -- 

2 


Therefore        fr».3™«co8arci(    ,,)2 
7C  2  (i  —  e2  sm20) 


If  we  now  put  for  brevity  -  —  -  —  -,  =  £,      ^J*      —  77  =  C, 

Rm  arc  i"  2  Ar#TO  arc  i" 

5  cos  a         LAO.    £  •    /  \\       j  *  +3tan20      ^   ,, 

-  77  =  h  (the  first  term  in  («)),  and  -  T2     —  -  =  E,  then 


arc  i 
Equa.  (n)  becomes 


+  (50'02  -D-h.s*-E-  sin2  a,     [64] 
and  the  new  latitude  is  given  by 

0'  =  0  +  A0".  [65] 

The  logarithms  of  the  factors  B,  C,  D,  and  £  are  given  in 
Table  XIV,  p.  351,  in  metric  units,  for  the  Clarke  spheroid  of 
1866. 

The  D  term  is  inserted  before  the  E  term,  because  it  is  usually 
the  larger.  The  E  term  may  be  omitted  when  log  s  is  less  than 
4.23.  .  .  .  The  D  term  may  be  omitted  when  log  s  is  less  than 
2.31  ...,  and  h2  may  be  substituted  for  (50")2  when  log  s  is  less 
than  4.93.  .  .  .  The  fourth  differential  coefficient  in  the  series 
may  be  neglected  except  for  the  very  longest  lines  (see  Coast  Sur- 
vey Report  for  1894,  p.  284) 

120.  Difference  in  Longitude. 

The  difference  in  longitude  is  such  a  small  angle  that  we  may 


DIFFERENCE  IN  LONGITUDE  165 

obtain  it  with  sufficient  precision  by  a  direct  solution  of  the  tri- 
angle PAB,  Fig.  70,  using  7-place  logarithms. 
Applying  the  law  of  sines, 

.    A  x       sin  o-  sin  a 

sin  AX  = : — 

cos  0 

The  sphere  on  which  the  points  are  projected  is  that  whose 
radius  is  N'  and  whose  center  is  at  Hf  corresponding  to  point  B. 

As  before,  let  a  =  —  • 

mi       e  •  .     s      sin  a  / .  x 

Therefore  sin  AX  =  sin-— 7 ;•  (p) 

N     cos  0 

In  practice  it  is  more  convenient  to  solve  the  equation  in  the 
form 

AX"  arc  i"  =  -^7  •  sin  a  sec  0", 

and  then  to  apply  corrections  for  the  difference  between  the  arc 
and  sine;  the  equation  should  therefore  be  written 

AX"  —  corr.iog  AX  =  TJ; 7,  •  sin  a  sec  0'  —  corr.iog  8, 

zV  arc  i 

since  each  side  of  the  equation  is  too  large  by  the  difference  be- 
tween the  arc  and  sine. 

Placing  TIT—   — —  =  A',  the  equation  becomes 
N  arc  i 

AX"  =  A  •  s  •  sin  a  sec  0'  +  corr.iogAx  —  corr.iogs          [66] 

in  which  the  corrections  are  to  be  applied  to  the  logarithms. 
Values  of  log  A'  will  be  found  in  Table  XIV,  p.  351. 
In  Art.  115,  p.  154,  it  was  shown  that 

,       5       .       .    s      Ms2 
log--logsin-  =  — > 
K  K       O  K* 

when  s  is  the  length  of  any  line  on  the  surface. 


1 66  CALCULATION  OF   GEODETIC  POSITIONS 


If  —  is  an  angle  expressed  in  seconds,  then  the  last  equation 
K 


becomes 


/c"\2 

M(S-}  arc2i" 
,      s      ,       .    s  \R/ 

log--logsm- 


R  R  6 

Taking  logs  of  both  members, 

i      tA-ct     ri      \       i      Mf  arc2 1  "\    ,      .     fs"\ 
log  (diff.  of  logs)  =  log   -  -    -f  2  log  —  • 

\        0        /  \K/ 

Applying  this  formula  first  to  AX", 

log  (diff.  of  logs)  =  8.2308  +  2  log  AX".  (q) 

Apply  the  formula  to  — ,  and,  observing  that  the  second  term  is 


log  (diff.  of  logs.)  =  8.2308  +  2  log  s  +  2  log  A'  (r) 

=   5. 2488*  +  2  log  S.  (s) 

This  correction  is  to  be  subtracted  because  arc  —7  is  greater  than 


sin&) 


In  Table  XIII  the  corrections  are  tabulated  to  show  the  values 
of  log  5  and  log  AA"  for  the  same  log  diff.  The  correction  for 
log  s  is  negative  and  that  for  log  A\"  is  positive.  The  algebraic 
sum  of  the  two  corrections  is  to  be  added  to  log  AXr/.  The 
method  of  making  these  corrections  is  illustrated  in  the  example 
on  p.  1 70.  The  new  longitude  X'  is  given  by 

V  =  X  +  AX".  [67] 

121.  Forward  and  Back  Azimuths. 

Owing  to  the  convergence  of  the  meridians  the  forward  and  re- 
verse azimuths  of  a  line  will  not  differ  by  exactly  180°,  as  in  plane 

*  Based  on  the  value  8.5090  for  log  A'. 


FORWARD   AND   BACK  AZIMUTHS  167 

coordinates.     The  amount  of  this  convergence  is  computed  as 
follows: 
In  the  triangle  PAB,  Fig.  70,  by  Napier's  analogies, 


2  COS  |  (7    +  T) 

Substituting,  and  noting  that  A  -f  B  +  Aa  =  180°,  and  that 
an  increase  in  AX  causes  a  decrease  in  A  a, 


2         sin  |0  +  *0 


whence  -  tan  i  A«  =  tan  i  AX 


2  2          COS  ^  (0  —  0 


2  A< 

cos  — 

2 


Therefore 


Putting  for  J  Aa  the  series 


.^^L\ 

A0 

cos  —  I 

2   / 


IAX.^l-irtaniAA.^T+  . 
»         COSM       3          2         cosA^ 

2   J  L  2   J 


and  for  tan  5  AX  the'  series 


then 


^T 
OSA0 

2  J 


1  ^    sin0m       AX3    sin0OT  ^    AX3    sin3<ftm 

2  A0       24  A</>       24          ,A0 
cos  -^              cos  —  -  cos3  —  - 

222 


1 68  CALCULATION  OF  GEODETIC  POSITIONS 

AX3 

Multiplying  by  2  and  factoring  out  — > 

24 


sin0OT        _i_  ,A  ,3  /  sin0m    _    sin3</)m  \ 
s  J  A<£      12  \cos     A</>      cos3 1  A0/ 


cos  ^  A0      12  Vcos  J  A</>      cos3 1  A0> 

Placing  cos  \  A0  =  i  in  the  small  term  and  reducing  A  a  and 
AX  to  seconds  -of  arc, 

-  A«"  =  AX"  coffe  +  ^ (AX")3  Sin  *•  C°S2  **  ^  '"         I 
=  AX"  sin  <t>m  sec  ^  +  (AX")3  -  F,  [68] 

in  which  F  is  an  abbreviation  for  ^  sin  <f>m  cos2  $m  arc2 1"  and  is 
given  by  its  log  in  Table  XIV.     This  F  term  amounts  to  only 
o".oi  when  log  AX"  =  3.36.  .  .  . 
The  back  azimuth  a  is  given  by 

a   =  a  +  A«  +  180°.  [69] 

In  calculating  the  geodetic  position  of  a  point,  the  azimuth  of 
the  line  to  that  point  is  to  be  found  from  the  known  azimuth  of 
the  fixed  side  of  the  triangle  by  using  the  corrected  spherical 
angle,  not  the  plane  angle  of  the  auxiliary  triangle.  The  com- 
putations of  0'  and  X'  may  be  verified  by  computing  the  position 
from  two  sides  of  the  triangle  and  noting  whether  the  same  $' 
and  X'  are  obtained  from  the  two  lines.  The  reverse  azimuths 
are  checked  by  noting  whether  their  difference  equals  the  spher- 
ical angle  at  the  new  station.  In  this  manner  the  calculation  of 
each  triangle  may  be  made  to  check  itself. 

122.  Formulae  for  Computation. 

For  convenience  of  reference  the  working  formulae  are  here 
brought  together. 

AX  =  A',  s.  sin  a  sec  0'  [66] 

*  The  value  of  —  A<£  may  be  made  more  accurate  by  the  addition  of  the  following 
term: 

-  \  s2  •  k  -  E  +  |  s2  cos2  a  •  k  •  E  +  \  s2  •  cos2  a  sec2</>  •  A2  -  k  arc2  i", 
in  which  k  =s2  •  sin2  a  •  C. 


FORMULA   FOR  COMPUTATION  169 

(or,  log  AX"  =  logs  +  CiogA>-Ciogs+log  sin  a  +  log  4'  +  logsec  0'), 
-  Ac*  =  AX"  sin  f(0  +  0')  sec  i  A0  +  (AX")3 .  F,  [68] 

in  which 

/?  =  s  •  cos  a  •  B, 
—  50  =  s  •  cos  a  •  5  +  s2  sin2  a  •  C  —  hs2  sin2  a  •  E. 

The  position  of  the  new  point  and  the  reverse  azimuth  are  then 
given  by 

0'  =  0  +  A0,  [65] 

X'  =  X  +  AX,  [67] 

a    =  a  +  Ac*  +  180°.  [69] 

The  arrangement  of  the  computation  is  illustrated  by  the  fol- 
lowing example.  The  two  pages  show  the  two  computations  of  a 
position  in  the  same  triangle. 

In  the  first  page  of  the  computation,  the  known  station  is  Waldo 
and  the  position  of  Bunker  Hill  is  to  be  found.  Since  the  value 
of  Aa  depends  upon  AX  and  AX  depends  upon  0',  the  three  parts 
of  the  solution  must  be  carried  out  in  the  order  indicated.  In 
computing  A0,  take  out  B,  C,  D,  and  E  for  the  given  latitude  0. 
The  (60)  used  in  the  D  term  is  usually  taken  as  the  algebraic  sum 
of  the  first  two  terms  of  the  series;  if  the  E  term  is  large,  it  should 
be  included  also.  The  h  in  the  E  term  is  the  first  (B)  term  alone. 
The  algebraic  signs  of  the  functions  of  a  are  important  and  should 
be  carefully  attended  to. 

When  computing  AX,  0'  is  known  and  the  factor  log  A'  must 
be  taken  out  for  this  new  latitude  0',  not  for  0.  The  primes  are 
inserted  to  call  attention  to  this.  To  correct  for  the  difference 
between  the  arc  and  the  sine,  enter  Table  XIII  with  log  AX  and 
logs  as  arguments.  The  algebraic  sum  of  the  two  values  of 
"log.  diff."  is  the  correction  to  be  applied  to  log  AX.  The  value 
of  Aa  is  found  last. 

The  values  of  0'  and  X'  are  checked  by  noting  whether  the 
same  values  are  obtained  from  the  two  computations.  The  two 
reverse  azimuths  should  differ  by  the  spherical  angle  at  the  new 
station,  which  checks  the  computations  of  Aa. 


i  yo 


CALCULATION  OF   GEODETIC   POSITIONS 


a 

Z. 
a 
Aa 

a' 

Waldo  to  Meade's  Ranch 
Meade's  Ranch  and  Bunker  Hill 
Waldo  to  Bunker  Hill 

255°    i?'  I?'1-  52 
86     20    54    .50 

341        38      12    .02 

+4    43  .09 

Bunker  Hill  to  Waldo 
Third  angle 

180° 
161      42    55  .11 
38      08    34  .02 

<i> 

A0 

*' 

39°     09'  55".  645 
-17  39   -209 

Waldo 
s  =  34,407  .  64  meters 
Bunker  Hill 

X 
AX 
X' 

98°      49'  So".  128 
-07  29    .652 

38       52  16   .436 

98       42  20    .476 

5 

cos  a 
B 
h 

ist  term 
2d  term 

3rd  and  4th 
terms 
-A* 

i  (0  +  *0 

4.5366549 
9-97730I8 
8.5109150 

5* 
sin2  a 
C 

3d  term 
4th  term 

5 

sin  a 

9-07331 
8.99674 
i  31553 

(60)2 

D 

Arg 

5 

AX 

6.0499 
2.3832 

-h 
sz  sin2  a 
E 

(AX)3 
F 

AX 

sin  J  (0  +  0') 
sec  i  (A0) 

-Aa 

3-  0249  n 
8.0700 
6.0871 

3.0248717 

1058".  9409 
o   .2429 

9.38558 

+0.0271 
—0.0015 

8.4331 

—21 

+03 

7  .  1-820  n 

7-959 
7.872 

1059   -1838 
+  -  0256 

+0.0256 
4.5366549 
9.  498  3680  n 

8.5091469 
0.1087088 

5.831 

2  .  652  877  n 
9-799043 

1059.2094 
39°  01'  06".  04 

A' 

sec  0' 

AX 

2.  652  8786  n 
18 

Corr. 

-18 

2.451  921  n 
-283".  09 

2.6528768  n 
449"  -652 

123.  The  Inverse  Problem. 

Not  infrequently  it  is  required  to  find  the  distance  and  mutual 
azimuths  between  two  stations  whose  latitudes  and  longitudes 
are  known. 

If  we  place  x  =  s  sin  a  and  y  =  s  cos  a,  then,  from  Equa.  [66] 
and  [64],  we  have 

AX  cos  <// 
* =     -r-  [70] 


and             y  =  —  - 

[A0  +  Cx2  +  D  (50)2  -f  E  (A0)  a8], 

[71] 

from  which 

a;      AX  cos  <// 

TITl    rv      •    •     ••              

[72] 

Lclll  Ci    -                                 . 

and 

5  =  y  sec  o:       ) 

[73] 

=  x  cosec  a.  S 

LOCATION  OF  BOUNDARIES 


171 


a 

Meade's  Ranch  to  Waldo 

75°      28'  14".  52 

a 

Bunker  Hill  and  Waldo 
Meade's  Ranch  to  Bunker  Hill 

55        30  33    -73 

19       57  40   -79 

Aa 

-06  n    .66 

180 

a* 

Bunker  Hill  to  Meade's  Ranch 

199       51  29   .13 

0 

39°     13'  26".  686 

Meade's  Ranch 

X 

98°     32'  30".  506 

0' 

—21    IO     .250 

s  =  41,661.11  meters 
Bunker  Hill 

AX 
X' 

+09  49    .969 

38       52   16   .436 

98        42   20    .475 

5 

4.6197308 

s2 

9  23946 

(60)' 

6.2076 

-A 

3-  1037  n 

cos  a 
B 
h 

9.9730924 
8.5109105 

sin2  a 
C 

9.0664^ 
1.31644 

D 

2.3835 

s2  sin2  a 
E 

8.3060 
6.0882 

3.1037337 

9-62239 

8.5911 

7-  4979  n 

istterm 

+1269.795 

3d  term 

+0.0390 

(AX)» 

8.312 

2d  term 

0.419 

4th  term 

—0.0031 

F 

7.871 

+1270.214 

+0.0359 

6.183 

3d  and  4th 

5 

4.6197308 

term 
-A0 

+0.036 

sin  a 
A' 

9  5332455 

8.509  1469 

Arg. 

-31 

AX 

sin  J  (0  +00 

2.770830 
9.799317 

+1270.250 

t*+« 

39°  02'  51".  56 

sec  0' 

o  1087088 

AX 

+06 

sec  J  (A0) 
-Aa 

2 

2.7708320 
-25 

Corr. 

-25 

2.570  149 
371".  66 

2.7708295 

AX 

+589".  9694 

The  inverse  solution  may  be  worked  out  on  the  same  printed 
form  that  is  used  for  the  direct  solution,  but  the  order  of  procedure 
is  modified  as  follows:  First,  compute  x  by  Equa.  [70],  then  the 
C,  D,  and  E  terms  in  Equa.  [71],  obtaining  finally  y.  The  azi- 
muth is  then  found  through  its  tangent;  the  calculation  of  5  is 
the  final  step. 

124.  Location  of  Boundaries. 

Whenever  it  becomes  necessary  to  establish  on  the  ground  a 
boundary  line  between  two  states  or  countries,  the  length  of  the 
lines  and  the  accuracy  demanded  usually  make  it  necessary  to 
employ  geodetic  methods.  A  boundary  may  consist  of  a 
meridian  arc,  a  parallel  of  latitude,  or  a  great  circle  inclined  to  the 
meridian;  or  it  may  be  a  combination  of  these. 


172  CALCULATION  OF   GEODETIC   POSITIONS 

125.  Location  of  Meridian. 

If  a  boundary  is  a  meridian  arc  the  longitude  of  which  is  fixed 
by  law,  it  is  first  necessary  to  assume  approximate  positions  for 
the  terminal  points,  and  then  to  determine  the  longitude  of  these 
by  direct  observations.  These  points  are  then  corrected  in 
position.  After  the  terminals  have  been  established  on  the 
ground,  the  line  may  be  run  from  one  to  the  other  as  a  random 
line,  to  be  subsequently  corrected  if  necessary.  Observations  on 
Polaris  for  azimuth  will  show  the  direction  of  the  meridian.  The 
line  is  then  run  out  by  backsighting  and  foresighting.  If  neces- 
sary, the  direction  of  the  meridian  may  be  determined  at  inter- 
mediate points.  When  the  second  point  is  reached,  the  error  in 
the  running  of  the  line  becomes  known,  and  the  random  line  may 
be  set  over  or  re-run  in  the  usual  manner.  If  the  boundary  is 
long,  the  intermediate  points  may  be  found  by  triangulation  in- 
stead of  by  direct  measurement.  In  any  case  triangulation  will 
furnish  a  valuable  check. 

126.  Location  of  Parallel  of  Latitude. 

In  order  to  establish  a  parallel  of  latitude  on  the  ground,  it  is 
necessary  to  assume  a  point  as  nearly  as  may  be  on  the  desired 
parallel.  The  exact  position  of  this  assumed  point  is  then  de- 
termined by  Talcott's  method,  and  the  station  moved,  if  neces- 
sary, to  the  correct  position.  If  the  difference  between  the  ob- 
served and  the  desired  latitude  is  A<£,  the  sea-level  distance 
which  the  station  must  be  moved  is  s'  =  Rm  A0"  •  arc  i". 

At  higher  elevations  s'  should  be  increased  in  proportion  to  the 
distance  from  the  center  of  the  earth  (Equa.  [6]).  If  the  error  in 
position  proves  to  be  large,  it  may  be  advisable  to  make  another 
determination  of  the  latitude,  in  order  to  avoid  the  effect  of 
station  errors.  (See  Art.  83,  p.  109). 

The  next  step  is  to  determine  the  azimuth  of  a  reference  mark, 
by  observation  on  Polaris,  and  to  establish  the  direction  of  a 
great  circle  at  right  angles  to  the  meridian  (prime  vertical). 
Points  on  the  parallel  are  then  determined  by  measuring  offsets 
from  the  prime  vertical  as  a  reference  line. 


LOCATION  OF  PARALLEL  OF  LATITUDE  173 

In  Fig.  71  we  have,  in  the  triangle  PAB, 

PA  =  90°  -  0, 

A  =90°, 

and  tan  <r  =  tan  AX  cos  <£, 

or  ff  =  tan"1  (tan  AX  cos  0). 


FIG.  71. 

Expanding  a  by  the  formula  for  tan"1  #,  p.  330,  and  also 
tan  AX  in  terms  of  AX  by  the  formula  for  tan  x,  p.  330,  we  have 

(7  =  AX  cos  0  +  i  (AX  cos  #)3  tan2  <j>, 
or  s  =  ffN  =  NAX"  •  cos  <£  •  arc  i" 

+  %N  (AX"  cos  0  •  arc  i")3  tan2  0,  [74] 

which  gives  the  distance  AB  corresponding  to  any  difference  in 
longitude  AX. 

If  in  Equa.  [64]  we  place  a  =  90°, 


'  2  #12.  arc  i" 

The  offset  P  from  the  prime  vertical  (tangent)  for  any  distance 
from  the  initial  point  is 

[75] 


174  CALCULATION  OF   GEODETIC   POSITIONS' 

Since  P  varies  as  s2,  the  offsets  for  equidistant  intervals  along 
the  line  may  be  readily  calculated.  The  direction  of  the  pole 
from  any  point  (x)  on  AB  is  given  by 

PxA  =  90°  +  Aa, 
in  which  it  is  sufficiently  accurate  to  take 

—  Aa  =  AX  sin  <j>m.  [76] 

Since  the  numerical  value  of  A  a  increases  directly  as  AX,  it  will 
be  sufficient  to  take  the  increments  of  A  a  as  proportional  to  s. 

If  the  arc  of  the  parallel  is  a  long  one,  it  is  advisable  to  break 
it  into  sections,  and  to  establish  a  new  point  at  the  beginning  of 
each  section  by  direct  latitude  observation. 

(See  United  States  Northern  Boundary  Survey,  Washington,, 

1878.) 

127.  Location  of  Arcs  of  Great  Circles. 

The  general  method  of  laying  out  arcs  not  coincident  with  the 
meridian  is  that  of  determining  astronomically  the  latitudes  and 
longitudes  of  the  terminal  points,  and  then  running  a  random 
line  between  them.  The  direction  and  distance  between  the 
terminals  may  be  found  by  Formulae  [70]  to  [73]  for  the  inverse 
solution  of  the  geodetic  problem.  The  azimuth  is  determined 
by  observation  at  intermediate  points.  The  error  of  the  random 
line  is  corrected  in  the  usual  way.  For  long  arcs  triangulation 
would  be  substituted  for  direct  measurement. 

(See  Appendix  3,  Coast  Survey  Report  for  1900,  "The  Oblique 
Boundary  Line  between  California  and  Nevada.") 

128.  Plane  Coordinate  Systems. 

When  all  the  points  to  be  located  in  a  survey  are  comprised 
within  a  relatively  small  area,  such  as  a  city  or  a  metropolitan 
district,  the  calculations  are  greatly  simplified  by  the  use  of  plane 
coordinates.  If  there  are  reliable  triangulation  points  already 
established  within  the  area,  these  will  naturally  be  used  as  a  basis 
for  the  new  survey,  or  at  any  rate  to  check  the  new  triangulation. 

In  establishing  a  system  of  plane  coordinates  it  is  necessary  to 
decide  first  upon  the  positions  of  the  coordinate  axes.  These 


CALCULATION  OF   PLANE   COORDINATES  175 

will  naturally  be  a  meridian  and  a  great  circle  at  right  angles  to 
it;  or,  more  properly  speaking,  they  will  be  straight  lines  tangent 
to  these  two  circles  at  their  point  of  intersection,  all  points  being 
supposed  to  lieHn  the  plane  denned  by  these  two  lines.  The 
origin  of  the  system  must  be  denned  in  terms  of  the  coordinates 
of  some  specified  point  of  the  survey  (geodetic  datum,  p.  158). 
Unless  this  is  done,  the  origin  will  not  be  the  same  when  derived 
from  different  points,  and  ambiguity  will  exist  regarding  the  true 
position  of  the  origin.  The  origin  may  be  taken  as  coincident 
with  the  selected  triangulation  point,  as  in  the  case  of  the  survey 
of  Boston,  Massachusetts,  and  Baltimore,  Maryland;  or  it  may 
be  the  intersection  of  a  selected  meridian  and  parallel  as  derived 
from  the  assigned  latitude  and  longitude  of  some  station.  In 
Springfield,  Massachusetts,  for  example,  the  origin  is  the  inter- 
section of  the  42°  04'  parallel  and  the  72°  28'  meridian,  as  de- 
termined by  the  published  latitude  and  longitude  of  the  United 
States  Armory  flagpole.  The  direction  of  the  meridian  must  be 
defined  as  making  a  certain  angle  with  a  specified  Jine  of  the  sur- 
vey, preferably  one  which  passes  through  the  fundamental  point. 

The  point  at  which  the  plane  is  tangent  to  the  spheroid  must 
not  be  confused  with  the  (o,  o)  point  of  the  system.  The  former 
should  be  within  the  area  surveyed,  preferably  at  its  center,  in 
order  to  avoid  large  spherical  errors.  The  latter  may  be  taken 
at  any  convenient  distance  outside  the  area  by  assigning  to  the 
tangent  point  large  values  of  x  and  y,  in  order  to  avoid  negative 
values  in  the  coordinates  of  the  survey  points.  The  tangent 
point  is  on  the  sphere  as  well  as  on  the  plane;  the  (o,  o)  point  is 
not  necessarily  on  the  sphere. 

129.  Calculation  of  Plane  Coordinates  from  Latitude  and 
Longitude. 

In  calculating  the  plane  coordinates  of  a  point,  we  may  apply 
Formulae  [70]  to  [73]  for  the  inverse  solution  of  the  geodetic 
problem,  one  of  the  points  being  the  origin  (tangent  point)  whose 
coordinates  are  <£  and  X,  and  the  other  the  triangulation  point  the 
coordinates  of  which  are  </>'  and  X'.  The  x  and  y  there  given  are 


I76 


CALCULATION  OF  GEODETIC  POSITIONS 


the  plane  coordinates  desired.  If  the  coordinates  of  many  points 
are  to  be  transformed,  it  will  prove  to  be  more  convenient  to 
use  specially  prepared  auxiliary  tables  and  to  modify  the  calcula- 
tions as  follows. 

In  Fig.  72  P  is  the  triangulation  point  whose  latitude  and 
longitude  are  known,  and  whose  coordinates  x  and  y  with  refer- 
ence to  the  origin  0  are  desired.  For  such  distances  as  are  likely 


FIG.  72. 

to  occur  in  a  plane  system  it  may  be  assumed  that  PE  =  PD; 
that  is,  x  equals  the  length  of  the  arc  of  the  parallel  PD.  The 
ordinate  y  =  PC  may  be  taken  as  PA  (the  difference  in  latitude) 
plus  BC*  (the  offset  from  great  circle  to  parallel).  From  For- 
mula [70], 

[77] 


x  =  PD  =  AX"  • 
If  x  is  to  be  expressed  in  feet, 

„     COS0' 


A' 


*  =  AX".  =  x  3.2808$.  [7g| 

(See  Table  A.) 

*  If  P  is  south  of  the  origin,  the  offset  must  be  subtracted. 


CALCULATION  OF  PLANE   COORDINATES 


I77 


TABLE  A.     VALUES  OF  LOG  +  0.515  9842* 

Distance  west  of  origin  in  feet  =  x  =  AX"  X  H 


Lat.  <t>'. 

LogH. 

Lat.  *'. 

LogH. 

P.  P. 

570 

572 

574 

576 

0      1     II 

42  10 

1.8768536 

0     /      II 

42  20 

I.87S  7103 

i 

19 

19 

19 

19 

2 

38 

38 

38 

38 

30 

7966 

30 

6530 

3 

57 

57 

57 

58 

4 

76 

76 

77 

77 

II 

7396 

21 

5957 

5 

95 

95 

96 

96 

30 

6825 

3° 

5383 

6 

114 

114 

H5 

"5 

7 

133 

J34 

134 

134 

12 

6255 

22 

4809 

8 

152 

153 

153 

154 

9 

171 

172 

172 

173 

30 

5684 

30 

4235 

10 

190 

191 

191 

192 

13 

5114 

23 

3661 

ii 

209 

2IO 

2IO 

211 

12 

228 

229 

230 

230 

3° 

4543 

30 

3086 

13 

247 

248 

249 

250 

14 

266 

267 

268 

269 

14 

397i 

24 

2512 

15 

285 

286 

287 

288 

30 

3400 

30 

1937 

16 

304 

305 

306 

307 

17 

323 

324 

325 

326 

15 

2828 

25 

1362 

!8 

342 

343 

344 

346 

19 

36i 

362 

364 

365 

30 

2256 

30 

0787 

20 

38o 

38i 

383 

384 

16 

1684 

26 

I  .875  O2I2 

21 

399 

400 

402 

403 

22 

418 

419 

42i 

422 

3° 

III2 

30 

1.8749636 

23 

437 

439 

440 

442 

24 

456 

458 

459 

46! 

17 

1.8/6  0541 

27 

9o6l 

25 

475 

477 

478 

480 

3° 

1.8759968 

30 

8485 

26 

494 

496 

497 

499 

27 

513 

5i5 

517 

5i8 

18 

9396 

28 

7910 

28 

532 

534 

536 

538 

29 

55i 

553 

555 

557 

30 

8823 

30 

7334 

30 

570 

572 

574 

576 

i9 

8250 

29 

6757 

30 

7677 

3° 

6181 

20 

1.875  7I«>3 

30 

1.8745604 

This  is  the  form  adopted  by  the  city  of  Springfield,  Mass.,  lor  its  coordinate  system. 


i78 


CALCULATION  OF  GEODETIC   POSITIONS 


TABLE   B.     VALUES  OF  0.515  9842  -  log  B 
Dist.  N.  of  Origin  in  Feet  =  A0"  X  K  +  x2 
Dist.  S.  of  Origin  in  feet  =  A<f>"  X  K  —  x2 


Lat. 

Log.  K. 

Lat. 

Log.  K. 

P.  P.,  Diff.  i'  =  12.8. 

42  10 

2  .OO5  2891 

42  2O 

2.005  3I09 

n 
I 

O 

n 

22 

5 

30 

2988 

30 

3116 

2 

0 

23 

5 

II 

2994 

21 

3122 

3 

I 

24 

5 

30 

3000 

3° 

3129 

4 

I 

25 

5 

12 

3006 

22 

3135 

5 

I 

26 

6 

30 

3013 

3° 

3141 

6 

I 

27 

6 

13 

3019 

23 

3U7 

7 

I 

28 

6 

30 

3026 

30 

3154 

8 

2 

29 

6 

14 

3032 

24 

3160 

9 

2 

3° 

3039 

30 

3167 

10 

2 

15 

3045 

25 

3i73 

ii 

2 

30 

3052 

30 

3180 

12 

3 

16 

3058 

26 

3186 

13 

3 

30 

3064 

30 

3i93 

14 

3 

i? 

3070 

27 

3i99 

15 

3 

30 

3077 

30 

3205 

16 

3 

18 

3083 

28 

3211 

17 

4 

3° 

3090 

3° 

3218 

18 

4 

i9 

3096 

29 

3224 

19 

4 

3° 

3103 

30 

3231 

20 

4 

20 

2.005  3109 

30 

'2.005  3237 

21 

4 

The  difference  in  latitude  PA  is  converted  into  feet  by  multi- 
.28o8{ 
B 

tan</> 


plying  A0"  by  -~a.     (Table  B .) 
The  offset  EC  (Formula  [75])  = 


A 


[79] 


The  factor  a"J ,  in  feet,  may  be  taken  from  Table  C  which 


was  calculated  by  the  formula 


[80] 


*  For  another  method  of  calculating  this  offset,  see  an  article  entitled  "  A  Method 
of  Transforming  Latitude  and  Longitude  into  Plane  Coordinates,"  by  Sturgis  H. 
Thorndike,  Journal  Boston  Society  Civil  Engineers,  Vol.  3,  No.  7,  September,  1916. 


CALCULATION  OF  PLANE  COORDINATES 


179 


TABLE  C.    VALUES  OF  LOG  (ft.)  =  log  C  -  log  B  -  0.515  9842 

Offset  from  parallel  =  log  L  +  2  log  x 


Lat. 

Log.  L. 

Lat. 

Log.L. 

P.  P.  Diff.  i'  =  25.4. 

o    /    // 

42  10 

2-33  46o 

0     /     // 

42  20 

2-33  7H 

I 

O 

24 

IO 

30 

473 

30 

727 

2 

I 

25 

II 

II 

486 

21 

739 

3 

I 

26 

II 

30 

499 

30 

752 

4 

2 

27 

II 

12 

S12 

22 

765 

5 

2 

28 

12 

30 

525 

3° 

778 

6 

3 

29 

12 

13 

537 

23 

790 

7 

3 

30 

55° 

30 

803 

8 

4 

14 

562 

24 

815 

9 

4 

3° 

575 

3° 

828 

10 

-  4 

15 

587 

25 

840 

ii 

5 

30 

600 

30 

853 

12 

5 

16 

612 

26 

865 

13 

6 

30 

625 

30 

878 

14 

6 

17 

638 

27 

892 

IS 

6 

3° 

651 

30 

905 

16 

7 

18 

663 

28 

917 

17 

7 

30 

676 

30 

930 

18 

8 

i9 

689 

29 

942 

i9 

8 

30 

702 

30 

955 

20 

8 

20 

2-33  7H 

3° 

2.33967 

21 

9 

22 

9 

23 

10 

Example.  As  an  illustration  of  how  this  method  would  be  applied,  let  us  sup- 
pose that  it  is  desired  to  compute  the  plane  coordinates  of  A  Powder  horn  in  a  system 
whose  origin  is  the  dome  of  the  State  House,  Boston,  Massachusetts.  We  first 
compute  A<£"  and  AX"  and  then  apply  formulae  [78],  [79]  and  [80]  as  shown. 

Powderhorn      Lat.  42°  24'  04".683         Long.  71°  01'  52".oo6 
State  House  42    21  29  .596  71   03  51  .040 


log  x2  =  7.9°l83 
logZ,  =  2.33752 


•"2o835"'°87 

logA<£"     =  2.190  5754 
\ogK  =  2.005  3I29 


AX 


log 

Offset 


0.23935 
1.7352  ft. 


log 


4.195  8883 

15699.59  ft. 
1-74 


59-034 

"  =  ii9".o34 

log  AX"  =  2.075  6710 
logZ7  =  1.875  2422 

log*  =  3.9509132 

x  =  8931.27  ft.  East  of 
State  House. 


y  =  15701.33  ft.   North  of  State  House 


If  it  is  preferred  to  make  the  conversion  from  AX  to  #  always 
on  the  same  parallel  of  latitude,  that  of  the  origin,  a  table  may  be 
calculated,  giving  the  length  of  each  minute  (i'  to  10')  and  each 


i8o 


CALCULATION  OF   GEODETIC  POSITIONS 


second  (i"  to  60")  of  arc  on  this  parallel;  the  difference  in  longi- 
tude may  be  taken  out,  by  parts,  from  this  table.  If  this  is  done, 
however,  it  is  necessary  to  make  allowance  for  the  convergence 
of  the  meridians  between, the  two  parallels  by  solving  for  the  dis- 
tance AB  =  y  sin  6  (Fig.  72).  The  convergence  6  =  AX"  sin  <f>m 
and  its  sine  may  be  tabulated  for  different  values  of  AX  and  $m. 
If  the  triangulation  point  is  north  of  the  origin,  AB  is  to  be  sub- 
tracted; if  south,  it  is  to  be  added. 

130.  Errors  of  a  Plane  System. 

.  In  order  to  investigate  the  errors  of  a  plane  coordinate  system 
like  the  preceding,  let  us  assume  that  a  line  starts  from  the  origin 
o,  Fig.  73,  in  an  azimuth  a,  and  follows  the  surface  of  a  sphere  of 


G 


0 
FIG.  73. 


radius  ^/RmN  (for  latitude  <j>)  for  a  distance  5  meters,  to  point  A ; 
and  that  another  line  OA' ',  having  the  same  azimuth  and  length, 
lies  in  the  plane  which  is  tangent  to  the  sphere  at  o.  The  point 
Af  in  the  plane  then  represents  the  point  A  on  the  sphere  as  de- 
termined by  a  direct  measurement  from  the  origin.  The  defects 
of  the  plane  system  as  a  means  of  representing  points  on  a  sphere 
will  be  shown  by  the  error  in  reproducing  point  A'  by  following 
different  routes,  such,  for  example,  as  traversing  due  north  and 
then  due  west  on  the  sphere,  or  due  west  and  then  due  north. 
If  a  perpendicular  AF  (an  arc  of  a  great  circle)  be  let  fall  from 


ERRORS  OF  A  PLANE   SYSTEM  181 

A  (Fig.  73)  to  the  meridian  through  o,  its  length  will  be  deter- 
mined by 

.a        .    s 
sin  —  =  sin  —  •  sin  a, 
R  R 

where  a  is  the  perpendicular  distance  in  meters  and  R  is  the 
radius  of  the  sphere. 

For  the  corresponding  distance  on  the  plane, 
a  =  s  •  sin  a. 

Distinguishing  the  plane  and  spherical  values  of  a  by  sub- 
scripts, p  and  s,  the  difference  in  length  may  be  found  as  follows : 

+ 

dp  —  a*  =  s  sin  a  —  Rsiif1  (sin  a  sin  — ) 

o|f^3 


Rssina  .     s3     .  s3     .  , 

=  5.sma___+_sma__sm3a+ . . . 

=  --—sin  a  cos2  a  -f-  •  •  •  . 
6  R 

Assuming  that  </>  =  40°,  a  =  N  45°  W,  and  s  =  20,000  meters 
(about  12  miles),  then  ap  —  a,  =  om.on6.  If  another  such  line 
were  to  extend  2o,ooom,  N  45°  E,  to  B,  the  terminal  points  A  and 
B  would  then  be  0*^.0232  farther  apart  if  calculated  on  a  plane 
than  if  calculated  on  the  sphere.* 

If  the  survey  proceeds  from  o  northward  to  the  point  F,  where 
the  great  circle  from  A,  perpendicular  to  the  meridian,  inter- 
sects that  meridian,  and  then  westward  along  this  great  circle  to 
A,  the  point  A  would  be  reached  without  error,  if  the  measure- 
ments were  perfect.  The  point  computed  on  the  plane  would 
not  agree,  however,  with  A'  as  already  established.  The  excess 
of  the  spherical  distance  b8,  along  the  meridian  to  the  foot  of  the 
perpendicular  F,  over  the  plane  distance  bp  is  found  as  follows: 

*  This  does  not  refer  to  the  chord-distance  AB,  but  to  the  distance  on  the 
spherical  surface. 


182 


CALCULATION  OF  GEODETIC   POSITIONS 


In  the  spherical  right  triangle, 


tan-=r  =  tan  §cosa. 
K.  K. 


Then 


b,  —  bp  =  R  tan  1  (tan  —  cos  a  J  —  s  cos  a 
s3  cos  a  sin2  a 


Assuming  the  same  data  as  before,  we  find  that  in  order  to  reach 
A,  on  the  sphere,  we  must  run  N  14142.15886  meters  and  then 
W  14142.12400  meters.  Since  in  this  case  5  sin  a  =  scosa  = 
14142.  135  63  w,  such  a  traverse,  when  computed  on  the  plane, 
gives  a  point  om.o2323  N  and  om.on63  E  of  point  A'  .  A  similar 
traverse  running  west  to  point  G  (Fig.  73)  and  then  north  to  A 
Would  give  a  point  om.on63  S  and  om.o2323  W  of  point  Ar.  The 
relative  positions  are  shown  (actual  size)  in  Fig.  74. 


A' 


From  O  direct 


From  O  north 
then  west 


FromO  west 
then  north 


FIG.  74. 


FIG.  75. 


The  maximum  discrepancy  in  the  traverse  is  then  about  ow.o5, 
or  about  two  inches.  This  would  appear  as  an  error  of  closure 
of  the  traverse  OF  AGO  even  if  there  were  no  error  whatsoever  in 
the  measurements  themselves. 

The  difference  in  length  between  an  arc  of  the  parallel  and  an 


ADJUSTING  TRAVERSES  TO  TRIANGULATION  183 

arc  of  the  great  circle  is  found  as  follows:  In  Fig.  75,  J  AB  = 
r  sin  —  =  R  sin  -  .     Replacing  the  sines  by  their  series  in  terms  of 

2  2 

the  arcs,  r  (  ---  )  =  R  (  ---  -V     The   difference  between 
\  2        48  /          \2      48; 

r  AX,  the  arc  of  the  parallel,  and  Re,  the  arc  of  the  great  circle,  is 


24  24 

AX3       D    AX3  cos3  0,  N 

=  R  cos  0  --  R  --     -  (approx.) 
24  24 

since  6  =  AX  cos  0,  nearly. 

Therefore         r  AX  -  R0  =  ^  R  cos  0  AX3  (i  -  cos2  0) 

=  2T  R  OX")3  .  arc3  i"  cos  0  sin2  0. 

In  order  to  compare  this  with  the  previous  examples,  we  must 
put  AX/r  =  ii92".4,  which  corresponds  to  the  distance  between 
A  and  B.  The  error  r  AX  —  RO  is  found  to  be  om.oi86  for  the 
total  arc,  or  0^.0093  for  the  half  arc.  The  difference  between 
the  length  of  the  parallel  and  the  x  coordinate  is  therefore 
om.on6  —  0^.0093  =  om.oo23. 

These  results  indicate  that  a  plane  system  may  be  extended 
over  an  area  twelve  miles  in  radius  without  involving  errors  of 
computation  as  great  as  the  errors  of  measurement,  and  also  that 
the  formulae  given  may  be  used  whenever  it  is  safe  to  use  plane 
coordinates. 

131.   Adjusting  Traverses  to  Triangulation. 

Whenever  a  traverse  is  to  be  run  from  one  triangulation  point 
to  another,  or  if  the  circuit  is  to  return  to  the  original  triangula- 
tion point,  some  method  must  be  provided  to  allow  for  the  effect 
of  convergence  of  the  meridians.  The  most  obvious  method  is 
to  refer  all  bearings  in  the  traverse  to  the  direction  of  the  initial 
meridian,  taking  no  account  of  true  bearings  at  any  other  point 
of  the  survey.  This  method  is  subject  to  very  small  errors,  far 
within  the  limit  of  accuracy  of  the  field  measurements,  unless  the 
area  is  much  greater  than  that  ordinarily  covered  by  a  traverse. 


184  CALCULATION  OF  GEODETIC   POSITIONS 


PROBLEMS 

Problem  i.  Calculate  the  latitude  and  longitude  of  point  A,  Problem  3,  Chapter 
VI,  from  both  lines,  and  the  back  azimuths  AB  and  AC. 

Problem  2.  Calculate  the  latitude  and  longitude  of  point  E,  Problem  4,  Chapter 
VI,  and  the  back  azimuths  EL  and  EN. 

Problem  3.    Calculate  the  portion  of  Sand  Hill  in  Problem  5,  Chapter  VI. 

Problem  4.  What  will  be  the  error  of  closure  of  a  survey  which  follows  the  cir- 
cumference of  a  circle  whose  radius  is  20,000  meters  (on  the  earth's  surface)  if  the 
survey  is  calculated  as  though  it  were  on  a  plane,  the  latitude  of  the  center  being 
40°  N.  and  the  measurements  being  exact? 


CHAPTER  VIII 
FIGURE  OF  THE  EARTH 

132.  Figure  of  the  Earth. 

The  term  "figure  of  the  earth  "  may  have  various  interpreta- 
tions, according  to  the  sense  in  which  it  is  employed  and  the  de- 
gree of  precision  with  which  we  intend  to  define  the  earth's 
figure.  When  we  say  that  the  earth  is  spherical,  we  mean  that 
the  sphere  is  a  rough  approximation  to  the  true  figure,  sufficiently 
close  for  many  purposes.  We  adopt  the  sphere  to  represent  this 
figure  because  it  is  a  simple  surface  to  deal  with  mathematically. 
When  a  closer  approximation  is  required,  we  employ  the  spheroid, 
or  ellipsoid  of  revolution.  This  figure  is  so  near  the  truth  that  no 
closer  approximation  has  ever  been  needed  in  practical  geodetic 
operations,  although  an  ellipsoid  (three  unequal  axes)  or  an 
ovaloid  (southern  hemisphere  the  larger)  may  be  nearer  the 
truth.  All  the  surfaces  mentioned  are  regular  mathematical  sur- 
faces, substituted  for  the  true  surface  on  account  of  their  sim- 
plicity. 

In  defining  the  true  figure  it  is  necessary  to  distinguish  be- 
tween the  topographical  surface  and  that  surface  to  which  the 
waters  of  the  earth  tend  to  conform  because  they  are  free  to 
adjust  themselves  perfectly  to  the  forces  acting  upon  them.  It 
is  this  latter  surface  with  which  we  are  chiefly  concerned  in 
geodesy;  the  land  surface  is  not  referred  to  except  in  such  ques- 
tions as  the  effect  of  topography  upon  the  direction  and  in- 
tensity of  gravity.  The  true  figure^  called  the  geoid,  is  defined 
as  a  surface  which  is  everywhere  normal  to  the  force  of  gravity, 
that  is,  an  equipotential  surface;  and  of  all  the  possible  surfaces 
of  this  class  it  is  that  particular  one  which  coincides  with  the 
mean  surface  of  the  oceans  of  the  earth.  Under  the  continents 

185 


i86 


FIGURE  OF  THFv  EARTH 


it  is  the  surface  to  which  the  waters  of  the  ocean  would  tend  to 
conform  if  allowed  to  flow  into  very  narrow  and  shallow  canals 
cut  through  the  land.  It  is  necessary  to  suppose  these  canals 
narrow  and  shallow  in  order  that  the  quantity  of  water  removed 
may  not  modify  the  figure  over  the  ocean  areas. 

Some  idea  of  the  relation  of  the  spheroid,  the  geoid,  and  topo- 
graphical surface  may  be  gained  by  an  inspection  of  Fig.  76.  It 
will  be  seen  that  the  geoidal  surface  coincides  with  the  surface 
of  the  ocean,  and  that  it  intersects  the  spheroid  at  some  distance 
out  from  the  shore  line.  The  inclination  of  the  normal  to  the 


FIG.  76. 

plumb  line  (station  error)  shows  the  angle  between  the  two  sur- 
faces at  this  point. 

The  surface  of  the  geoid  may  be  represented  conveniently  by 
means  of  contour  lines  referred  to  the  spheroid  as  a  datum  sur- 
face. In  Fig.  77,  which  shows  contours  of  the  geoid  within  the 
limits  of  the  United  States  proper,  that  portion  of  the  contours 
shown  in  full  lines  is  taken  from  a  map  published  by  the  Coast 
and  Geodetic  Survey  in  "  Figure  of  the  Earth  and  Isostasy  " 
(1909);  the  remaining  portions  (dotted)  were  sketched  in  by 
eye,  following  in  a  general  way  the  topography  of  the  continent. 
Such  a  map  conveys  no  real  information  about  the  elevations 
of  the  geoid  except  along  the  full  lines,  but  is  given  simply  to 
show  how  the  contours  would  be  used  in  representing  the  geoid. 

When  we  speak  of  the  spheroid  as  the  ''figure  of  the  earth  "  we 


DIMENSIONS  OF  THE  SPHEROID   FROM  TWO  ARCS        187 

mean  that  particular  spheroid  which  best  represents  the  earth  as 
a  whole,  or  which  most  closely  fits  some  specified  area.  The 
dimensions  of  such  a  spheroid  are  not  to  be  regarded  as  fixed, 
but  are  subject  to  revision  with  each  accession  of  new  data.  Such 
a  spheroid  necessarily  depends  upon  a  large  amount  of  data,  and 
the  calculations  for  fixing  its  dimensions  are  long  and  compli- 
cated, involving  the  adjustment  of  many  observations  by  the 
method  of  least  squares. 


127°     122°   117°    112°   107°   102"    97"      »"     8T     82°      77°      72  3       67° 


FIG.  77.     Contours  of  the  Geoid. 

The  principal  methods  of  determining  the  spheroid  are  (i)  by 
the  measurement  of  arcs,  which  may  be  portions  of  meridians, 
of  parallels,  or  of  great  circles;  (2)  by  means  of  areas  containing 
several  astronomical  stations  rigidly  connected  by  triangulation ; 
and  (3)  by  observations  of  the  force  of  gravity. 

133.  Dimensions  of  the  Spheroid  from  Two  Arcs. 

The  simplest  method  by  which  the  dimensions  of  the  spheroid 
can  be  determined  is  by  the  measurement  of  two  meridian  arcs. 
The  length  of  each  arc  and  the  latitudes  of  the  terminal  points  of 
each  must  be  measured.  If  the  earth  were  a  perfect  spheroid, 
and  if  there  were  no  errors  of  measurement,  the  two  arcs  would 
determine  exactly  the  elements  pf  the  spheroid. 


i88 


FIGURE  OF  THE  EARTH 


In  the  equation  of  the  ellipse  there  are  two  constants  to  be 
determined,  and  it  will  be  shown  that  the  determination  of  the 
curvature  of  the  meridian  ellipse  at  two  points  will  enable  us  to 
compute  these  constants  and  consequently  all  the  other  elements 
of  the  ellipse.  In  Fig.  78,  suppose  that  the  lengths  of  the  two 


FIG.  78. 

meridian  arcs  have  been  measured  by  triangulation  and  that 
their  lengths  are  s  and  s',  and  that  the  differences  of  the  latitudes 
of  their  terminals  are  A<£  and  A0',  respectively.  The  radii  of 
curvature  of  the  ellipse  at  the  middle  points  of  the  arcs  are 

a  (i  -  e2) 


and 


Rm  = 


Rm'  = 


a  (i  -  i2) 


in  which  <£  and  <£'  refer  to  the  middle  points  of  the  arcs  and  a 
and  e  are  unknown.  If  the  two  arcs  are  regarded  as  arcs  of 
circles  whose  radii  are  to  be  found,  then 


and    RJ  = 


arc  i 


arc  i 


DIMENSIONS  OF  THE  SPHEROID   FROM  TWO  ARCS       189 

are  the  two  radii  of  curvature,  A<£  being  in  seconds.  The 
shorter  the  arcs,  the  less  the  error  involved  in  assuming  that 
they  are  circular. 

Equating  the  two  values  of  Rm  and  Rmf ,  we  have 

s  a(i  -e?)  ,  , 


s'  a  (i  - 

A7^=(l-^si 

Dividing  (a)  by  (b)  and  solving  for  e2, 


Having  found  e2  from  Equa.  [81],  the  equatorial  radius  a  may 
be  computed  by  substituting  the  value  of  e2  in  either  (a)  or  (6). 
The  value  of  b  may  then  be  found  from  the  relation 

62  =  a2  (i  -  e2).  (c) 

The  compression  /  is  given  by 

r  d    —    b  r         i 

/  -  —  [531 

The  length  of  a  quadrant  of  the  meridian  'may  be  found  by 
applying  Equa.  [54],  Chapter  V. 

In  this  method  of  determining  the  elements  of  the  spheroid  it 
should  be  observed  that  there  are  just  enough  measurements  to 
enable  us  to  solve  the  equations,  and  no  more.  All  errors  of 
measurement  enter  the  result  directly;  we  should  not,  therefore, 
expect  to  derive  very  accurate  values  from  two  arcs. 

As  an  illustration  of  the  preceding  method  let  us  take  the 
Peruvian  Arc  and  a  portion  of  the  Russian  Arc,  the  data  for 
which  are  as  follows: 


190 


FIGURE  OF  THE  EARTH 


PERUVIAN  ARC 


Station. 

Astr.  lat. 

Dist.  in  meters 
between  the  parallels 
of  latitude. 

Tarqui 

o           /                 // 

S  3  04  32  068      } 

Cotchesqui 

N  o  02  •*!  387      \ 

344,740.5 

RUSSIAI 

$  ARC  (Northern  End) 

Tornea 

N    6^    AQ    A.A.    <7         } 

Fuglenaes 

tl  u->  4-y  44-i>/       c 

N    7O   AO      II    23        1 

539»84L7 

Substituting  in  Formulae  81,  (a)  and  (c),  the  resulting  values 


are 


e2  =  0.0065473, 


b  —  6,356,440  m. 

134.   Oblique  Arcs. 

If  an  arc  (ABj  Fig.  79)  is  inclined  to 
the  meridian  at  a  small  angle,  it  may  be 
utilized  to  determine  the  curvature  of  the 
meridian  as  follows:  Referring  to  Equa. 
(n),  Chapter  VII,  it  is  seen  that  the  dif- 
ference in  latitude  of  the  terminal  points 
of  the  line  is  given  by  the  series  for  A0". 
Hence  the  length  of  the  meridian  arc  is 
given  by  A0".  Rm  •  arc  i",  and 

A0"  •  Rm  •  arc  i"  =  —  s  cos  a —  s2  sin2  a  tan  0 

2  N 

+  — —  s3  sin2  a  cos  a  (i  +  3  tan2  0) .      [82] 

Each  line  of  a  chain  of  triangles  may  be  projected  onto  the  meri- 
dian, and  its  length  found  by  this  formula.  The  length  and  dif- 
ference in  latitude  of  the  end  points  are  thus  found,  and  the 
projection  treated  as  though  it  were  a  measured  meridian  arc. 


FIG.  79. 


FIGURE  OF  THE  EARTH  FROM   SEVERAL  ARCS  IQI 

The  sum  of  all  these  short  arcs  may  then  be  treated  as  a  single 
arc  to  be  combined  with  another  similar  arc  in  the  computation 
of  a  and  e. 

135.  Figure  of  the  Earth  from  Several  Arcs. 

When  several  arcs  are  to  be  used  to  determine  the  elements  of 
the  spheroid,  there  are  more  data  than  are  necessary  for  the 
direct  solution  as  given  in  Art.  133.  The  arcs  usually  consist  of 
several  sections;  that  is,  the  latitudes  of  several  stations  along 
the  same  meridian  are  observed  and  the  distances  between  them 
are  determined  by  the  triangulation.  The  problem  is  one  of 
combining  all  these  measurements  by  the  method  of  least  squares 
in  order  to  obtain  the  most  probable  values  of  the  elements. 
Only  the  outline  of  the  method  can  be  given  here. 

From  Equa.  [49]  we  have  for  the  length  of  a  meridian  arc 

s  =  A0  •  Rm  •  arc  i", 

which  is  sufficiently  accurate  for  short  arcs.  For  long  arcs  a 
more  accurate  expression  is  necessary.  Suppose  that  an  arc 
consists  of  several  sections,  the  latitude  of  the  initial  point  being 
0i,  the  second  02,  etc.,  and  that  the  meridian  distances  between 
the  stations  are  s,  si,  etc.  From  the  first  two  latitudes 


Instead  of  finding  a  and  e*  directly,  it  is  more  convenient  to 
assume  approximate  values  of  these  quantities  and  to  compute 
the  most  probable  corrections.  Let  us  assume  the  equations 

a  =  a0  +  5# 
and  e2  =  e?  +  5e*. 

Let  Ro  be  the  value  of  Rm  corresponding  to  e<?  and  a0.    Ex- 
panding (f)  by  Taylor's  theorem, 


IQ2  FIGURE  OF  THE  EARTH 

Evaluating  the  two  differential  coefficients, 

d(^\ 

\Rm/  _        (i  —  g2sin20)g      £ 
~ftiT  a2  (i  -  e2)      ~  tf' 

neglecting  the  e2  terms,  and 

•d(*\ 

\R/  _    _  a  (i  —  e2)  •  f  « (i  —  e2  sin2 0)2  sin2 0  —  (i  —  e2  sin20)^  *a 
de2  .  a2  (i  -  e2) 

=  -  (i  —  f  sin2  0),  neglecting  e2  terms. 
a 

Substituting  these  values  in  (g), 
Hence  (e)  becomes 


The  errors  in  the  measured  latitudes  are  so  large  in  comparison 
with  the  errors  in  the  measured  arcs  that  the  lengths  are  con- 
sidered exact  and  the  observed  latitudes  are  given  corrections 
Vi,  %,  etc.  Equa.  (ti)  then  becomes 


In  the  small  terms,  containing  da  and  8e2,  the  e2  terms  were 
omitted;  that  is,  e2  was  placed  equal  to  zero.    This  makes  Rm  =  a 

and  02  -  <#>i  =  -      —r,  in  these  terms. 


Substituting  in  (i), 


.        x  q         .       9         v       ^ 

+  (i-  f  sin20)- 


S 


PRINCIPAL  DETERMINATIONS  OF  THE  SPHEROID         193 

If  we  place 

x  =  da, 

y  =  d<?,       . 
substituting  in  (/'),  we  have 

a&  +  biy  +  h  =  %  -  »i,  (£) 

where  a,  =  -  ^A 


It  is  evident  that  an  equation  of  this  form  (k)  may  be  written 
for  each  section  of  each  arc.  There  will  be  more  equations  than 
there  are  unknown  quantities  to  be  found.  From  these  equations 
we  may  form  a  set  of  "  normal  "  equations  (Art.  201,  p.  293), 
equal  in  number  to  the  number  of  unknown  quantities,  that  is, 
equal  to  the  number  of  arcs  plus  two.  The  simultaneous  solu- 
tion of  the  normal  equations  gives  the  corrections  da  and  Se2,  and 
also  the  correction  to  the  initial  latitude  of  each  arc. 
136.  Principal  Determinations  of  the  Spheroid.* 
The  spheroids  which  have  been  most  extensively  used  are  those 
of  Bessel  (1841)  and  Clarke  (1866).  Bessel's  determination  was 
based  on  the  following  arcs;  the  Peruvian,  French,  English, 
Hannoverian,  Danish,  Prussian,  Russian,  Swedish,  and  two 
Indian  arcs.  The  resulting  elements  of  the  spheroid  are  generally 
used  in  Europe  at  the  present  time  in  geodetic  surveys.  They 
were  employed  in  the  United  States  up  to  about  1880.  Clarke's 
spheroid  (1866)  was  calculated  from  the  following  six  arcs,  the 
total  amplitude  being  about  76°  35';  the  French,  English, 
Russian,  South  African,  Indian,  and  the  Peruvian.  The  Clarke 
spheroid  is  larger  and  flatter  than  Bessel's.  It  was  adopted  by 
the  Coast  and  Geodetic  Survey  about  1880,  after  it  became  evi- 
dent that  the  surface  in  this  part  of  the  globe  has  a  flatter  curva- 

*  For  an  account  of  the  different  arc  measurements  see  A  History  of  the  Determi- 
nation of  the  Figure  of  the  Earth  from  Arc  Measurements,  by  A.  D.  Butterfield, 
Worcester,  1906. 


194 


FIGURE   OF   THE   EARTH 


GEODETIC  DATUM  195 

ture  than  that  indicated  by  the  Bessel  spheroid.  The  semiaxes 
of  these  two  spheroids  are  shown  below,  their  dimensions  being 
based  on  Clarke's  value  of  the  meter,  namely,  im  =  39.370113 
inches. 


a  (meters). 

b  (meters). 

Bessel  (1841).    .  .                        . 

6  377  ^Q7 

6  356  079 

Clarke  (1866).                             .      . 

6  378  206 

6  356  584 

Several  other  spheroids  have  been  calculated  from  different 
groups  of  arcs,  but  have  not  been  extensively  used  Tor  geodetic 
purposes. 

137.   Geodetic  Datum. 

The  question  of  where  to  place  the  spheroid  with  respect  to  the 
station  points  of  a  survey,  and  the  question  whether  a  certain 
spheroid  properly  represents  the  curvature  of  the  area  being 
surveyed,  are  determined  by  a  comparison  of  the  geodetic  and 
astronomical  positions  of  the  survey  points.  As  the  survey 
progresses  the  geodetic  latitudes  and  longitudes  will  be  calculated 
on  the  surface  of  the  adopted  spheroid,  starting  from  some 
assumed  position  of  one  of  the  triangulation  stations.  At  the 
same  time  the  positions  of  many  of  the  stations  will  be  deter- 
mined astronomically.  The  differences  in  the  latitudes,  as- 
tronomical minus  geodetic  (A  —  G),  the  differences. in  the  longi- 
tudes, and  the  differences  in  the  azimuths  are  computed  for  every 
station  where  the  astronomical  observations  have  been  made. 
A  study  of  these  differences  and  their  manner  of  distribution  will 
show  what  corrections  to  the  assumed  position  of  the  initial 
point  will  reduce  the  algebraic  sum  of  the  quantities  (A  —  G)  to 
a  minimum.  If  these  differences  were  due  wholly  to  errors  in 
the  assumed  latitude  and  longitude  of  the  initial  point,  it  would 
be  possible  to  reduce  ^  (A  —  G)  to  zero,  but  a  part  of  this  differ- 
ence is  due  to  local  deflection  of  the  vertical,  that  is,  to  the  dif- 
ference in  slope  of  the  geoidal  and  spheroidal  surfaces.  For  this 


196  FIGURE  OF  THE  EARTH 

reason  the  most  that  can  be  expected  is  to  place  the  spheroid  so 
as  to  reduce  ^  (A  —  G)  to  a  small  quantity.  The  remaining 
values  of  (^4  —  G)  at  the  different  stations  after  a  recomputation 
has  been  made,  serve  to  indicate  the  slope  of  the  geoid  with 
reference  to  the  spheroid. 

If  the  reference  spheroid  adopted  has  too  great  a  curvature, 
the  computed  latitudes  will  increase  or  decrease  faster  than  the 
astronomical  latitudes  as  the  survey  proceeds  north  or  south 
from  the  initial  point  (Fig.  80).  This  was  observed  as  the  sur- 
veys in  this  country  were  gradually  extended  on  the  Bessel 


FIG.  80. 

spheroid.  If  we  consider  an  area  instead  of  a  meridian  arc,  then 
we  see  that  if  all  the  astronomical  zeniths  are  swung  inward  with 
reference  to  the  geodetic  zeniths,  the  spheroid  that  we  are  using 
for  the  calculations  must  have  too  great  a  curvature  for  the  area 
in  question.  If  the  observed  latitudes  are  sometimes  too  great, 
sometimes  too  small,  as  we  proceed  along  a  meridian,  this  simply 
shows  that  the  verticals  are  deflected  locally,  and  that  the 
average  curvature  of  the  surface  is  nearly  that  of  the  spheroid. 

138.   Determination  of  the  Geoid. 

The  form  of  the  geoid  is  determined  by  observing  the  local 
variations  from  the  spheroid  as  a  surface  of  reference.  These 
deviations  may  be  determined  either  from  the  station  error 


EFFECT  OF  MASSES  OF  TOPOGRAPHY 


197 


(difference  between  astronomical  and  observed  position)  or  from 
the  observed  variation  in  the  force  of  gravity. 

The  station  error  at  any  point,  or  local  deflection  of  the  vertical, 
is  a  direct  measure  of  the  slope  of  the  surface  of  the  geoid  with 
reference  to  the  spheroid.  The  geodetic  coordinates  of  the  point 
are  computed  with  reference  to  a  line  normal  to  the  spheroid, 
while  the  astronomical  coordinates  are  referred  to  the  actual 
direction  of  the  plumb  line,  which  is  normal  to  the  geoidal 
surface. 

139.  Effect  of  Masses  of  Topography  on  the  Direction  of  the 
Plumb  Line. 

The  deflection  of  the  plumb  line  by  masses  of  topography  may 
be  computed  by  applying  Newton's  law  of  gravitation,  that  is, 
if  mi  and  m^  be  any  two  masses,  D  the  distance  between  them,  and 
k  a  constant  (to  be  found  by  experi- 
ment), then  the  force  of  attraction 
between  mi  and  nh  is 


Station 


that  is,  the  force  of  attraction  is  pro- 
portional to  the  product  of  the  masses 
and  varies  inversely  as  the  square  of 
the  distance  between  them.  The 
effect  of  any  mass,  such  as  a  moun- 
tain, in  deflecting  the  direction  of  YIG.  81. 
gravity  at  any  station  may  be  found 

by  combining  the  attraction  of  the  mountain  with  the  attraction 
of  the  earth  regarded  as  a  sphere.  It  may  be  shown  that  the 
attraction  of  a  sphere  at  any  external  point  is  the  same  as  though 
its  mass  were  concentrated  at  its  center.  The  relative  attrac- 
tions of  the  mountain  ana  --he  earth  upon  the  plumb  bob  at  the 

station  are  as  —  to  —  (Fig.  81),  where  m  is  the  mass  of  the  moun- 
tain, M  that  of  the  earth,  and  d  the  distance  of  the  mountain 


FIGURE  OF  THE  EARTH 


from  the  station.     The  angle  D  through  which  the  plumb-line  is 
deflected  is  given  by 


tanD  = 


mR2 
Md2 


The  earth's  mass  is  f  wR3  X  5.58  (the  constant  5.58  being  the 
mean  density  of  the  earth).  If  the  mountain  has  a  volume  v  and 
density  5,  and  the  earth's  radius  be  taken  as  6370  kilometers, 
then 

0'  =  0.00138 1,  [83] 

the  dimensions  being  in  meters  and  the  angle  in  seconds. 

N 


FIG.  82. 

In  order  to  take  into  account  all  of  the  topography  about  a 
station  when  computing  the  deflection  of  the  plumb  line,  the 
following  method  may  be  employed  (see  Clarke,  Geodesy,  p.  294). 
The  area  surrounding  the  station  is  supposed  to  be  divided  into 
circular  rings  of  any  desired  width,  and  the  rings  cut  into  four- 
sided  compartments  by  radial  lines,  as  in  Fig.  82. 

It  is  desirable  to  separate  the  component  of  the  deflection  in 


EFFECT  OF  MASSES  OF  TOPOGRAPHY  199 

the  meridian  plane  from  that  in  the  prime  vertical.  Let  h  be  the 
height  of  the  upper  surface  of  the  mass  above  station  0;  let  a  and 
r  be  the  azimuth  and  horizontal  distance  to  any  particle  P  in  the 
mass;  and  let  z  be  its  height  above  0  and  6  its  density.  The  mass 
of  the  particle  is  then  6  •  r  •  da  •  dr  •  dz.  The  attraction  of  the 
particle  on  0  is 

,     5  •  r  •  da  •  dr  •  dz 
r*+* 

k  being  the  gravitation  constant.* 

The  component  of  this  attraction  in  the  plane  of  the  meridian 
is  the  total  attraction  multiplied  by  the  cosine  of  the  angle  be- 
tween PO  and  50,  which  is 


Vr2  +  z2 

The  total  attraction  of  the  mass  in  the  compartment  in  the 
direction  50  is 

dr-dz 


_rzdrdz 

ri  «/o    ' 

dr 


=k  r  r  ch8'r2- 

JaiJri  Jo 

=  k  •  5  (sin  a   —  sin  ai)    I       / 
Jri  Jo 

=  k  •  5  •  h  (sin  a'  —  sin  <*i)   I 

Jri 

=  k  •  8  •  h  (sin  a  —  sin  «i)  log« 


Unless  h  is  very  large,  "the  equation  may  be  written    with 
sufficient  accuracy 

A  =  kdh  (sin  a  —  sin  ai)  log*  — ; 


Vrt+h* 

r'  +  VT^ 


that  is,  the  mass  is  considered  to  lie  in  the  plane  of  the  horizon 
of  the  station. 

*  The  gravitation  constant  may  be  defined  as  the  attraction  of  one  unit  mass  on 
another  unit  mass  at  a  unit  distance  away.  In  the  C.  G.  S.  system  this  is 
6673  X  io-u. 


200 


FIGURE  OF  THE  EARTH 


The  attraction  of  the  earth  at  point  0,  supposing  it  to  be  a 
sphere  of  radius  R  (3960  miles)  and  of  density  A,  is 


~ 


The  angle  of  deflection  in  the  plane  of  the  meridian  is  given 
by  the  ratio  of  attractions,  that  is, 

h  (sin  a  —  sin  <*i)  loge  — 
n  _  5  ri 


=  1 2" . 44  —  •  h  •  (sin  a   —  sin  «i)  loge  — 
A  r\ 


[84] 


The  ratio  of  densities  —  may  be  taken  as ;*  d  —  2.67  and 

A  2.09 


By  extending  the  rings  outward  this  computation  may  be 
carried  as  far  from  the  station  as  desired.  If  a  compartment  is 
very  far  from  the  station,  it  becomes  necessary  to  correct  for  the 
curvature  of  the  earth,  because  the  mass  no  longer  lies  in  the 
horizon  of  the  station,  as  at  first  assumed. 
*  See  Harkness,  The  Solar  Parallax  and  its  Related  Constants,  Washington,  1891. 


LAPLACE  POINTS  2OI 

If  the  angles  «i  and  a  are  measured  from  the  prime  vertical 
instead  of  from  the  meridian,  the  formula  gives  the  deflection  in 
a  plane  at  right  angles  to  the  meridian. 

By  the  foregoing  process  we  may  compute  for  any  station 
what  is  called  the  topographic  deflection.  It  shows  what  the 
deflection  of  the  plumb  line  would  be  if  no  other  forces  acted 
upon  it  than  those  mentioned.  A  comparison  of  the  values  so 
computed  with  the  station  errors  actually  observed  shows  the 
former  to  be  much  larger  than  the  latter;  from  which  we  infer 
that  the  attraction  of  the  surface  topography  cannot  be  the  only 
force  tending  to  deflect  the  plumb  line. 

Laplace  Points. 

As  stated  above,  it  is  customary  to  resolve  the  deflection  of  the 
plumb  line  into  two  components,  one  in  the  plane  of  the  meridian 
and  the  other  in  the  plane  of  the  prime  vertical.  The  meridian 
component  is  found  directly  by  subtracting  the  geodetic  (com- 
puted) latitude  from  the  observed  astronomic  latitude.  The 
prime  vertical  component  must  be  obtained  indirectly  either 
from  the  astronomic  and  geodetic  longitudes  or  from  the  astro- 
nomic and  geodetic  azimuths.  In  terms  of  the  longitudes  this 
component  is 

p.  v.  component  =  (X^  —  XG)  cos  <fo. 

In  terms  of  the  azimuth  it  is 

p.  v.  component  =  —  (o^  —  a<?)  cot  <£#. 

Both  of  these  relations  may  be  derived  from  the  figure  (82a). 
If  we  equate  the  two  values  for  the  prime  vertical  component 
we  obtain 

(a A  —  «e)  =  —  (X^  -  Xo)  sin  4>G 

which  is  known  as  the  Laplace  equation.  Triangulation  stations 
at  which  the  astronomic  longitude  and  azimuth  have  been  ob- 
served are  called  Laplace  points. 

The  geodetic  and  astronomic  longitudes  in  the  United  States 
are  subject  to  probable  errors  of  less  than  o".5.  The  astronomic 
azimuths  are  also  determined  with  about  the  same  accuracy. 


202  FIGURE  OF  THE  EARTH 

The  geodetic  azimuths,  however,  as  carried  through  the  tri- 
angulation,  are  subject  to  an  error  about  ten  times  as  great.  The 
triangulation  may  therefore  be  greatly  strengthened  by  correcting 
the  geodetic  azimuths  at  Laplace  points  by  means  of  the  above 
equation. 

The  manner  of  correcting  the  geodetic  azimuth  is  illustrated 
by  the  following  example,  taken  from  Supplementary  Investiga- 
tion in  igog  of  the  Figure  of  the  Earth  and  Isostasy. 

U.  S.  Standard  longitude  of  Parkersburg      \   \  =    88°  01'  49^.00 

Astronomic                 "         _  "            "  =    88  01    48  .30 

A  —  G  in  longitude  —  o".7o 

A  —  G  in  azimuth  =  (—0.70)  (— sin</>0)  =  +o  .44 

Astronomic  azimuth  Parkersburg  to  Denver  =  143°  16    15  .55 

True  geodetic  azimuth  Parkersburg  to  Denver  =143  16    15. n 

U.  S.  Standard  azimuth  Parkersburg  to  Denver  =  143  16    15  .64 

Correction  to  U.  S.  Standard  azimuth  =  ~o"-53 

140.  Isostasy  —  Isostatic  Compensation. 

For  many  years  it  has  been  known  that  the  estimated  and 
observed  values  of  the  station  error  are  not  in  even  approximate 
agreement,  and  it  has  long  been  suspected  that  the  explanation 
would  be  found  in  the  fact  that  the  densities  of  the  material 
immediately  beneath  the  surface  are  unequal,  regions  of  deficient 
density  lying  beneath  mountain  ranges,  and  regions  of  excessive 
density  lying  beneath  low  areas  and  under  the  ocean  bottom.  It 
is  supposed  that  at  some  depth  the  excess  above  the  surface  is 
compensated  by  the  defect  below  the  surface,  and  vice  versa. 
This  condition  is  given  the  name  isostasy.  It  appears  that  the 
theory  was  first  clearly  stated  by  Major  C.  E.  Button  in  1889, 
and  since  that  time  it  has  been  the  subject  of  much  study. 

In  1909  and  1910  there  were  published  by  the  Coast  and  Geodetic 
Survey  the  results  of  a  very  extensive  investigation  conducted  by 
Professor  J.  F.  Hayford,  then  Inspector  of  Geodetic  Work  and 
Chief  of  the  Computing  Division.  The  investigation  was  based 
primarily  upon  the  computation  of  the  topographic  deflections 
at  a  large  number  of  astronomical  stations  in  the  United  States. 
The  best  topographic  maps  available-  were  used  for  this  purpose. 


ISOSTASY  — ISOSTATIC  COMPENSATION  203 

These  computed  deflections  were  then  compared  with  the  known 
(observed)  deflections  at  these  same  stations  as  found  from  the 
triangulation  and  astronomical  observations.  In  substantially 
all  cases  the  computed  deflection  was  found  to  exceed  the  ob- 
served deflection  by  a  large  amount,  although  the  two  were 
usually  of  the  same  algebraic  sign.  Computations  were  then 
made  to  test  the  theory  that  this  condition  called  isostasy  actually 
exists. 

The  condition  known  as  isostasy  may  be  stated  as  follows:  the 
mass  in  any  prismatic  column  which  has  for  its  base  a  unit  area  of 
the  horizontal  surface  lying  at  the  depth  of  compensation,  for  its 
edges  vertical  lines  (lines  of  gravity),  and  for  its  upper  limit  the 
actual  irregular  surface  of  the  earth  (or  the  sea  surface  if  the  area 
in  question  is  beneath  the  ocean) ,  is  the  same  as  the  mass  in  any 
other  similar  prismatic  column  having  a  unit  area  on  the  same 
surface  for  its  base.  Such  prismatic  columns  have  different 
heights  but  the  same  mass,  and  their  bases  are  at  the  same  depth 
below  the  geoidal  (sea-level)  surface. 

Computations  were  made  assuming  different  depths  of  com- 
pensation, for  the  purpose  of  finding  at  what  depth  the  computed 
deflections  (taking  isostasy  into  account)  most  nearly  agree  with 
the  observed  deflection.  It  was  found  that  the  compensation 
was  most  nearly  complete  (more  than  -£$  complete)  at  a  depth  of 
about  122  kilometers,  or  about  76  miles. 

It  should  be  observed  that,  while  the  densities  in  the  prismatic 
columns  tend  to  compensate,  the  resultant  deflection  of  the  plumb 
line  is  not  zero,  for  the  portions  of  the  column  nearest  the  station 
have  a  much  greater  influence  than  the  distant  portions.  The 
tendency  is  to  throw  all  the  zeniths  outward  from  the  continental 
dome,  assigning  to  the  surface  a  curvature  which  is  greater  than 
it  should  be.  Thus,  if  isostasy  is  not  taken  into  account,  the 
dimensions  of  a  spheroid  computed  from  such  data  will  be  too 
small.  This  investigation  not  only  included  a  determination  of 
the  most  probable  depth  of  compensation,  and  a  substantial 
proof  of  the  validity  of  the  theory  in  so  far  as  it  applies  to  the 


204 


FIGURE  OF  THE  EARTH 


United  States,  but  also  included  a  determination  of  the  most 
probable  dimensions  of  the  spheroid  for  that  area.  In  this  calcu- 
lation the  area  method  was  employed.  The  dimensions  of  the 
spheroid  resulting  from  this  investigation  are  as  follows: 

«  =  6,378,388™  ±  i8m, 
b  =  6,356,909™, 

j  =  297.0  ±  0.5. 

The  general  conclusions  in  regard  to  the  existence  of  isostasy 
within  the  limits  of  the  United  States  were  later  confirmed  by  the 
results  of  a  similar  investigation  of  the  compensating  effect  upon 
observed  values  of  the  force  of  gravity  determined  with  the 
pendulum. 

The  results  of  these  investigations  will  be  found  in  the  follow- 
ing publications  of  the  United  States  Coast  Survey: 

John  F.  Hayford,  The  Figure  of  the  Earth  and  Isostasy  from 
Measurements  in  the  United  States,  1909. 

John  F.  Hayford,  Supplementary  Investigations  in  1909  of  the 
Figure  of  the  Earth  and  Isostasy,  1910. 

John  F.  Hayford  and  William  Bowie,  The  Effect  of  Topography 
and  Isostatic  Compensation  upon  the  Intensity  of  Gravity,  Special 
Publication  No.  10,  1912. 

William  Bowie,  The  Effect  of  Topography  and  Isostatic  Com- 
pensation upon  the  Intensity  of  Gravity,  Special  Publication  No.  12, 
1912. 

William  Bowie,  Investigation  of  Gravity  and  Isostasy,  Special 
Publication  No.  40,  1917. 

PROBLEMS 

Problem  i.    Compute  the  dimensions  of  the  spheroid  from  the  following  arcs. 


Name. 

Lat.  of  middle 
point. 

Amplitude. 

Length  in  feet. 

Peruvian  (Delambre's)  
English                              

S       i    31  oo 

N      <<2    3<J    A.< 

3  07  03-1 
3^7    j  -2    i 

I  131  057 
I  44.2  Q<J3 

PROBLEMS  205 

Problem  2.    Compute  the  dimensions  of  the  spheroid  from  the  following  arcs. 


Station. 

Latitude. 

Distance  in  feet. 

Formentera 

0              /            // 

•?8   3Q    S3    17      ) 

Dunkirk 

ci   O2   08  41       \ 

4509  790.84 

Tarqui   .                                  ... 

S       3   O4.    32   O7       J 

Cotchesqui  ... 

:rT    *     ^  J       7     J^ 
IN    o  02  31  39     ) 

I  131  036.3 

Problem  3.  Lake  Superior  arc;  latitudes,  38°  43' i7//.22  and  48°  07'  o6".62; 
dist.,  1,043,974  meters.  Peruvian  arc;  latitudes,  — 3°O4/32//.o,  -fo°o2'3i."4; 
dist.,  344,736.8  meters.  Compute  a  and  e2. 


CHAPTER  IX 
GRAVITY  MEASUREMENTS 

141  .  Determination  of  Earth's  Figure  by  Gravity  Observations. 

The  determination  of  the  force  of  gravity  by  means  of  pendu- 
lums affords  a  second  means  of  determining  the  earth's  figure, 
which  is  entirely  independent  of  the  arc  method  previously  dis- 
cussed. In  this  method  the  force  of  gravity  is  measured  at  points 
of  known  latitude  and  longitude.  From  the  observed  variation 
of  gravity  with  the  latitude  the  polar  compression  may  be  com- 
puted. Such  measurements,  therefore,  will  give  the  form  but 
not  the  absolute  dimensions  of  the  spheroid. 

In  the  following  discussion  the  term  gravity  (g)  will  be  taken 
to  mean  the  resultant  obtained  by  combining  the  force  of  the 
earth's  attraction  due  to  gravitation  and  the  centrifugal  force 
due  to  the  rotation  of  the  earth. 

142.  Law  of  the  Pendulum. 

The  relation  between  /,  the  length  of  a  simple  pendulum,  P, 
its  period  of  oscillation,  and  g,  the  force  of  gravity  is  given  by  the 
formula 


or,  more  accurately, 


where  h  is  the  height  through  which  the  point  of  oscillation  falls 
during  a  half  oscillation. 

143.  Relative  and  Absolute  Determinations. 

Determinations  of  gravity  are  of  two  kinds: 

(i)  Absolute  determinations,  in  which  both  P  and  /  are  measured 
and  from  which  g  may  be  calculated;  and  (2)  relative  determina- 

206 


VARIATION  OF   GRAVITY  WITH  THE  LATITUDE  207 

lions,  in  which  P  is  measured  at  two  stations  and  the  ratio  of  the 
corresponding  values  of  g  at  the  two  places  becomes  known.  If 
the  time  of  oscillation  P  of  the  same  pendulum  has  been  ob- 
served at  two  stations,  then 

/7T2 


. 

and  &=^i» 

Ff 

whenoe  .fi.-  (87] 


Absolute  determinations  of  g  are  far  more  difficult  than  relative 
determinations,  owing  to  the  practical  difficulties  of  measuring 
the  length  /  with  sufficient  accuracy. 

Relative  determinations  may  be  made  with  very  great  ac- 
curacy, since  the  time  of  oscillation  may  be  measured  in  such  a 
manner  that  the  personal  errors  of  the  observer  have  but  little 
effect  on  the  results. 

Most  of  the  pendulum  observations  for  geodetic  purposes  are 
now  made  by  the  relative  method,  and  all  values  of  g  are  made 
to  depend  upon  some  one  reliable  determination  of  the  absolute 
value.  The  relative  values  of  g  in  such  a  system,  however,  still 
remain  more  accurate  than  the  computed  absolute  values. 

144.  Variation  of  Gravity  with  the  Latitude. 

The  approximate  law  governing  the  variation  of  gravity  with 
the  latitude  may  be  expressed  thus: 

[88] 

in  which  go,  ge,  and  gp  are  values  of  g  at  latitude  <fo,  at  the  equator, 
and  at  the  pole,  respectively.  By  means  of  two  such  equations, 
one  for  g+  observed  near  the  equator  and  one  for  g+  near  the  pole, 
the  two  unknowns  ge  and  gp  may  be  found. 

Equation  [88]  may  be  derived  in  a  simple  manner  if  we  may 


208 


GRAVITY  MEASUREMENTS 


neglect  variations  in  the  attraction  at  different  parts  of  the  surface.* 
Suppose  the  earth  to  be  a  sphere  of  radius  ry  the  attraction  G 
having  the  same  value  everywhere.  Then  g^  the  resultant  of 
the  attraction  G  and  the  centrifugal  force  cy  is  found  as  follows: 
At  the  equator  the  centrifugal  force  =  ce  =  wV.f  At  the  pole 
c  =  o. 


Also  at  the  equator 


ge  =  G  -  Ce 


and  at  the  pole 
whence 


gp  ~  ge   =  Ce 


FIG.  83. 


In  latitude  <£  (Fig.  83)  x  =  r  cos  0  and  c+  =  coV  cos  <f>  =  ce  cos  <£. 
The  component  of  C+  directly  opposed  to  G  is  ce  cos2  0  (vertically 
upward). 

Hence  #0  =  G  —  ce  cos2  0.  [89] 

*  See  Jordan's  Handbmh  der  Vermessungskunde,  Vol.  Ill,  p.  627. 

7)2 

t  The  centrifugal  force  may  be  expressed  by  -  ,  where  v  is  the  velocity  of  a  par- 
ticle at  the  equator.  The  distance  moved  by  a  particle  in  one  rotation  ( =  i  sidereal 
day  =  T  seconds)  is  zirr.  Hence  the  centrifugal  force  =  (  ^r)  r  =  coV,  where  &> 

is  the  angular  velocity.  T  =  86,400  sidereal  seconds  =  86,164.09  mean  solar 
seconds. 


VARIATION  OF  GRAVITY  WITH  THE  LATITUDE 


209 


Substituting  in  [89]  the  value  of  G  at  the  equator, 

g*  =  ge  +  ce  -  ce  cos2  <f> 
=  ge  +  ce  sin2  <f> 
=  go  +  (gP  -  £e)sin20; 

that  is,  g0  =  ge  (i  +  g-^L^      2 

\  ge 


sn 


[88] 


In  order  to  obtain  an  accurate  numerical  expression  for  g^, 
of  the  same  general  form  as  the  above,  we  may  write 

fr  =  £.(i+£  sin2  0) 

and  then  determine  the  value  of  B  which  is  in  best  agreement 
with  all  observed  values  of  g.  For  such  a  formula  Dr.  Helmert  * 
published,  in  1884,  the  equation 

go  =  978.0x30  (i  +  0.005310  sin2  <£),  [90] 

in  which  go  is  supposed  to  be  the  value  at  sea-level  and  the  unit 
is  dynes  of  force,  or  centimeters  of  acceleration. 

This  may  be  expressed  for  convenience  in  terms  of  go  at  lati- 
tude 45°.     Since  sin2  45°  =  J, 


and  since 
and 


2  sin2  0  =  i  —  cos  2  <j>, 
sin2  <j>  =  \  —  J  cos  2  < 


=  #45 


B 


2 


which  becomes 


go  =  980.597  (i  —  0.002648  cos  2  0). 
*  Helmert,  Hohere  Geodasie,Vo\.  II,  p.  241. 


[91] 


210  GRAVITY  MEASUREMENTS 

In  1901  Dr.  Helmert  gave  the  more  accurate  forms 

go  =  978.046  (i  +  0.005302  sin2  0  —  0.000007  sin2  2  </>)    [92] 
and  go  =  980.632  (i  —  0.002644  cos  2  </>  +  0.000007  cos2  2  </>),  [93] 

in  which  the  number  0.000007  (=  i-^0  ^s  a  coefficient  found 
theoretically  from  assumptions  regarding  the  internal  structure 
of  the  earth. 

These  formulae  refer  to  the  absolute  value  of  g  at  Vienna.  To 
refer  to  the  "  Potsdam  system/'  to  which  all  values  of  g  observed 
in  the  United  States  are  referred,*  the  equations  must  be  written 

g0  =  978.030  (i  +  0.005302  sin2  0  —  0.000007  si"2  2  0)    [94] 
and    go  =  980.616  (i  —0.002644  cos  2  ^+0.000007  cos2  2  0)-  [95] 

In  the  Coast  Survey  Special  Publication  No.  1 2,  entitled  "  Effect 
of  Topography  and  Isostatic  Compensation  upon  the  Intensity 
of  Gravity  "  (second  paper)  the  following  formula  is  given: 

go  =  978.038  (i  +  0.005302  sin2  0  —  0.000007  sin2  2  0)>  [96] 
equivalent  to 

go  =  980.624  (i  —  0.002644  cos  20+  0.000007  cos2  2  0)> 

which  is  Helmert's  formula  of  1901  corrected  by  0.008  dyne. 
The  constants  in  these  equations  were  derived  from  observations 
in  the  United  States  only. 

In  Special  Publication  No.  40,  a  study  is  made  of  observations 
in  the  United  States,  Canada,  Europe  and  India.  The  formula 
resulting  from  this  investigation  is 

go  =  978-°39  (x  +  0.005294  sin2  0  -  0.000007  sin2  0)>      [97] 

145.   Clairaut's  Theorem. 

The  relation  between  the  flattening  of  the  spheroid  at  the  poles 

*  The  American  observations  for  g  were  referred  to  Greenwich  (England),  Paris 
(France),  and  Potsdam  (Germany)  by  observations  made  in  1900  by  G.  R.  Putnam, 
(see  Coast  Survey  Report  for  1901). 


PENDULUM    APPARATUS  211 

and  the  values  of  gp  and  ge  is  expressed  by  Clairaut's  theorem, 
published  in  1743,  namely, 


in  which  ce  is  the  centrifugal  force  at  the  equator.  In  this  formula 
the  terms  of  the  second  order  have  been  omitted.  If  these  terms 
are  included,  the  formula  becomes 


=    ._B__..B__B       [98a] 

0  2     ge  \  3   W         T4     ge  21         21        / 

in  which  £  and  £4  are  coefficients  to  be  determined  from  the 
observations  (Helmert,  Hohere  Geodasie,  Vol.  II,  p.  83).  It  is 
by  means  of  this  equation  that  the  form  of  the  earth  is  com- 
puted from  gravity  observations. 

146.   Pendulum  Apparatus. 

Nearly  all  of  the  observations  of  gravity  for  geodetic  purposes 
are  made  with  pendulums  of  invariable  length,  by  the  relative 
method.  The  description  of  apparatus  in  the  following  articles 
will  be  limited  to  one  type,  the  half  -seconds  invariable  pendulum 
apparatus  as  designed  and  constructed  by  the  United  States 
Coast  Survey.  The  first  half-seconds  invariable  pendulum  with 
electrical  apparatus  for  determining  the  period  appears  to  have 
been  devised  by  Sterneck  (Austria)  in  1882.  In  1890  T.  C. 
Mendenhall,  then  Superintendent  of  the  Coast  and  Geodetic 
Survey,  designed  an  apparatus  of  this  kind  but  differing  in  many 
details,  however,  from  any  previous  design.  This  apparatus  has 
been  used  ever  since  that  time  in  substantially  the  same  form 
excepting  the  addition  of  the  interferometer  for  determining  the 
flexure.  This  apparatus  includes  three  half-second  pendulums, 
each  about  248""*  long  and  having  an  agate  plane  at  the  point  of 
suspension.  The  agate  plane  rests  on  a  knife-edge  support  (angle 
of  130°)  attached  to  the  pendulum  case  in  which  the  pendulums 
are  enclosed  when  they  are  swung.  The  use  of  the  blunt  angle 
on  the  knife  edge  and  the  placing  of  the  plane  (rather  than  the 


212 


GRAVITY  MEASUREMENTS 


PENDULUM   APPARATUS 


213 


knife  edge)  on  the  pendulum  are  designed  to  secure  greater 
permanence  of  length,  upon  which  the  accuracy  of  the  method 
depends.  The  pendulums  are  made  of  an  alloy  of  copper  and 
aluminum  and  weigh  1 200  grams  each.  The  three  are  of  slightly 
.different  lengths  so  that  they  will  have  different  periods.  Their 


FIG.  85.     Dummy  Pendulum  (with  thermometer),  Regular  Pendulum,  and 
Leveling  Pendulum. 
(C.  L.  Berger  and  Sons.) 

form  (Fig.  85)  is  such  as  to  give  strength  and  at  the  same  time 
offer  but  little  resistance  to  the  air.  In  addition  to  the  three 
observing  pendulums  there  is  a  dummy  pendulum,  of  the  same 
size  and  shape  but  carrying  a  thermometer  packed  in  filings  of 
the  same  metal.  There  is  also  a  small  pendulum  provided  with 
a  spirit  level  for  leveling  the  knife  edge.  Pendulums  made  of 


214 


GRAVITY  MEASUREMENTS 


invar  metal  are  now  (1919)  being  constructed  by  the  instrument 
division  of  the  Coast  and  Geodetic  Survey  so  that  it  will  be 
possible  to  make  gravity  observations  on  mountain  peaks  and 
other  places  where  the  control  of  temperature  is  difficult.  The 
use  of  this  metal  will  make  it  unnecessary  to  construct  a  "  con- 
stant temperature  room." 

The  pendulums  are  swung  in  an  air-tight  case  from  which  the 
air  may  be  nearly  exhausted  by  means  of  a  pump.  Levers  are 
provided  for  lowering  the  pendulum  onto  the  knife  edge  and  for 


FIG.  86.     Flash  Apparatus. 

starting  and  stopping  the  pendulum.  Inside  the  case  is  a  manom- 
eter tube  for  registering  the  air  pressure,  and  also  an  additional 
thermometer.  Levels  are  provided  for  leveling  the  case,  and 
there  is  a  graduated  scale  under  the  pendulum  for  reading  the  arc 
of  oscillation.  In  the  most  recent  work  of  the  Coast  Survey 
the  pendulum  receiver  has  been  enclosed  in  a  felt  and  leather 
case  to  prevent  fluctuations  in  temperature. 

The  observations  are  made  by  comparing  the  times  of  oscilla- 
tion of  the  pendulums  with  the  half-second  beats  of  a  break- 
circuit  (sidereal)  chronometer  connected  electrically  with  the 
"flash  apparatus  "  used  for  observing  the  coincidence. 


PENDULUM  APPARATUS  215 

The  flash  apparatus  (Fig.  86)  consists  of  a  shutter  a  operated 
by  the  armature  of  an  electromagnet  b  in  the  circuit  and  a  mirror 
c  behind  the  shutter  which  reflects  light  through  the  slit  d  to  two 
small  mirrors  e,  which  reflect  it  into  an  observing  telescope/;  one 
of  the  small  mirrors  is  attached  to  the  pendulum  and  the  other 
to  the  knife-edge  support.  In  the  most  recent  form  of  the 
flash  apparatus,  the  observer  looks  down  through  a  vertical  tel- 
escope and  sees  the  flash  reflected  by  a  prism.  This  arrange- 
ment is  more  convenient  for  the  observer  than  the  older  form 
because  the  pendulum  receiver  is  usually  mounted  on  a  very  low 
support. 

When  the  pendulum  is  at  rest  and  the  shutter  open,  a  beam 
of  light  from  a  lamp  *  at  one  side  of  the  apparatus  strikes  the 
mirror  c  at  an  angle  of  45°  and  passes  through  the  slit;  it  is 
reflected  from  both  mirrors  at  e  and  appears  to  the  observer  as 
two  horizontal  bright  slits  side  by  side.  The  mirrors  may  be 
adjusted  so  that  these  slits  appear  to  be  at  the  same  height,  so 
as  to  form  one  continuous  band.  If  the  pendulum  is  set  swing- 
ing, the  reflected  image  now  appears  to  travel  up  and  down, 
while  the  image  from  the  other  mirror  is  stationary.  If  the 
shutter  is  closed  and  allowed  to  open  only  for  an  instant  at  the 
end  of  each  second  (or  each  two  seconds),  the  observer  sees  that 
at  each  successive  opening  of  the  shutter  the  moving  image  has 
changed  its  position  relative  to  the  fixed  image.  This  is  due  to 
the  fact  that  the  period  of  the  pendulum  is  longer  than  the 
sidereal  second  and  the  pendulum  has  made  .slightly  less  than 
one  complete  (double)  oscillation.  By  watching  the  flashes  and 
noting  the  chronometer  readings  when  they  coincide,  the  ob- 
server obtains  the  number  of  seconds  between  two  successive 
coincidences.  During  this  interval  the  pendulum  has  evidently 
lost  just  one  oscillation  on  the  (half -second)  beats  of  the  chronom- 
eter. In  the  interval  between  two  successive  coincidences  the 
pendulum  has  made  one  less  than  twice  as  many  oscillations  as 

*  An  electric  bulb  placed  inside  the  flash  box  is  now  used  instead  of  the  oil 
lamp. 


2l6  GRAVITY  MEASUREMENTS 

the  chronometer  has  beat  seconds.  During  the  interval  between 
any  two  coincidences  the  number  of  oscillations  is  twice  the 
number  of  seconds  (s)  less  the  number  of  coincidence  intervals 
(n).  Hence  the  time  of  one  oscillation  (P)  is  given  by 


An  examination  of  this  formula  will  show  that  an  error  in 
noting  the  times  of  coincidence  produces  a  relatively  small  error 
in  P,  and  for  this  reason  the  method  is  almost  independent  of  the 
observer's  errors. 

On  account  of  the  variation  of  g  (and  consequently  of  P)  with 
the  latitude  of  the  station,  it  is  necessary  to  use  a  mean-time 
chronometer  at  stations  situated  near  the  pole,  because  the  period 
of  the  pendulum  approaches  so  closely  to  the  sidereal  half-second 
that  the  coincidence  intervals  are  inconveniently  long.  In  case 
a  mean-  time  chronometer  is  used,  the  formula  becomes 

[100] 


2  s  +  n 

147.  Apparatus  for  Determining  Flexure  of  Support. 

Observations  with  pendulums  mounted  on  a  very  flexible  sup- 
port show  plainly  that  when  a  pendulum  is  set  swinging,  it  com- 
municates motion  to  the  case  and  the  support  and  sets  them 
oscillating,  and  this  oscillation  in  turn  affects  the  observed  period 
of  the  pendulum.  The  apparatus  now  used  to  measure  the 
effect  of  this  flexure  is  one  which  operates  on  the  principle  of  the 
interferometer.*  This  is  an  optical  device  (Fig.  87)  consisting 
of  a  lamp  and  lens  arranged  so  as  to  furnish  a  beam  of  sodium 
light;  a  glass  plate  arranged  so  as  to  separate  the  beam  of  light 
into  two  parts,  one  of  which  is  transmitted,  the  other  reflected; 
two  mirrors,  one  in  the  path  of  each  beam  of  light;  and  a  telescope 
for  observing  the  image.  When  the  different  parts  of  the  appara- 

*  A  description  of  the  interferometer  will  be  found  in  the  Coast  Survey  Report 
for  1910. 


CORRECTIONS 


217 


'§  8 

Q  -8 


>  ^ 


2l8 


GRAVITY  MEASUREMENTS 


tus  are  properly  adjusted,  dark  and  light  bands  will  appear  in  the 
field  of  the  telescope,  owing  to  interference  of  the  sodium-light 
waves  of  the  two  beams.  One  of  the  mirrors  is  mounted  on  the 
pendulum  receiver,  while  the  rest  of  the  apparatus  is  on  an  inde- 
pendent support  in  front  of  it.  When  the  pendulum  is  set 
swinging,  it  sets  the  case  in  motion,  and  this  in  turn  moves  the 
mirror,  causing  a  slight  variation  in  the  length  of  the  path  of  one 
of  the  beams  of  light.  This  causes  the  interference  bands  to 
shift  back  and  forth;  the  amount  of  shift  may  be  estimated  by 
observing  the  motion  of  the  bands  over  a  cross-hair  or  a  scale  in 
the  field  of  the  telescope.  It  is  usually  observed  by  noting  the 


FIG.  88. 


scale  readings  of  both  edges  of  some  band  in  each  of  its  two  posi- 
tions (before  and  after  shifting).  The  movement  of  the  edges 
of  a  band  divided  by  the  width  of  the  band  (in  scale  divisions) 
gives  the  movement  in  units  of  the  width  of  a  band.  Fig.  88 
represents  the  interference  (dark)  bands  and  the  scale  divisions 
in  the  field  of  the  telescope. 

Tests  made  with  the  pendulum  mounted  on  supports  of  dif- 
ferent degrees  of  flexibility  will  show  the  relation  between  the 
observed  movement  of  the  fringe  bands  and  the  resulting  error  in 
the  period  of  the  pendulum.  In  the  Coast  Survey  tests  the  re- 
sults showed  that  a  movement  equal  to  the  width  of  one  band 
produced  a  change  of  173  in  P  in  units  of  the  seventh  decimal 
place.  This  is  more  conveniently  expressed  as  follows:  o.oi  F 


METHODS  OF  OBSERVING  2IQ 

produces  a  change  of  1.73  in  P,  where  F  is  the  width  of  a  band. 
This  constant  was  determined  with  the  pendulum  swinging 
through  an  arc  of  5mm  on  the  scale,  and  all  observed  flexures  must 
be  reduced  to  this  arc  before  correcting  P. 

148.  Methods  of  Observing. 

The  receiver  should  be  mounted  on  a  solid  support  such  as  a 
cement  or  brick  pier,  the  foot  screws  cemented  to  the  pier,  and 
the  instrument  sheltered  as  in  case  of  astronomical  observations. 
It  is  important  that  the  instrument  should  be  so  sheltered  that 
the  temperature  will  not  fluctuate  rapidly.  The  apparatus 
should  be  leveled  by  means  of  the  spirit  level  on  the  outside  of  the 
case  and  then  the  knife  edge  should  be  leveled  by  means  of  the 
leveling  pendulum.  In  moving  the  pendulums  great  care  should 
be  used  to  protect  them  from  injury  and  to  prevent  any  foreign 
matter  from  adhering  to  them.  The  accuracy  of  the  results  will 
depend  upon  the  permanency  of  length,  and  any  injury  due  to 
fall,  or  change  of  period  due  to  change  in  the  mass,  will  affect  the 
period  and  vitiate  the  results.  The  pendulums  should  not  be 
touched  with  the  hands,  but  should  be  lifted  by  means  of  a 
special  hook  made  for  this  purpose.  The  flash  apparatus,  chro- 
nometer, and  interferometer  should  be  placed  upon  supports  that 
are  entirely  independent  of  the  pendulum  support. 

Various  programs  of  observing  have  been  tried,  but  the  follow- 
ing has  been  chiefly  used  by  observers  of  the  Coast  Survey.  Each 
of  the  three  pendulums  is  swung  first  in  the  direct  and  then  in  the 
reversed  position,  making  six  swings  each  of  eight  hours'  dura- 
tion. The  error  of  the  chronometer  is  obtained  by  star-transit 
observations  (Arts.  52-71)  made  just  before  the  beginning  and 
at  the  end  of  the  series.  The  following  table  will  indicate  more 
clearly  the  order  of  operations. 

Star  Observations  9-10  P.M. 

Start  Pendulum  No.  i  10  P.M. 

Reverse  No.  i  6  A.M. 

Start  No.  2  2  P.M. 

Reverse  No.  2  10  P.M. 

Start  No.  3  6  A.M. 

Reverse  No.  3  2  P.M. 

Star  Observations  9  P.M. 
Stop  Pendulum  No.  3    after  star  observations 


220  GRAVITY  MEASUREMENTS 

If  star  observations  are  lost  at  the  end  of  the  set,  the  swings 
are  continued  until  star  observations  are  obtained.  At  the  begin- 
ning and  end  of  each  swing  several  coincidences  are  observed.  At 
the  end  of  each  swing  several  more  are  observed.  Very  little 
time  is  lost  between  swings,  so  that  they  are  almost  continuous 
between  star  observations.  For  this  reason  the  variations  in  the 
rate  of  the  chronometer  are  almost  entirely  eliminated  from  the 
mean  result  of  all  the  swings. 

Since  1913  the  Coast  Survey  observers  have  obtained  the 
chronometer  corrections  from  the  Naval  Observatory  time  sig- 
nals instead  of  by  direct  observations.  This  results  in  a  great 
saving  of  tune  and  cost.  Another  change  in  the  regular  pro- 
gram, recently  introduced,  is  to  swing  the  pendulums  for  twelve 
hours  instead  of  eight,  and  in  the  direct  position  only,  instead  of 
direct  and  reversed. 

After  a  pendulum  is  placed  in  position  on  its  support,  the  case 
closed,  and  the  air  exhausted  until  the  pressure  is  about  6omm,  the 
observer  lowers  the  pendulum  until  it  rests  upon  the  knife  edge, 
starts  it  swinging  through  an  arc  of  about  o°  53',  and  notes  the 
arc  on  the  scale.  To  observe  coincidences,  the  observer  switches 
in  the  chronometer  and  the  flash  apparatus  and  then  watches  the 
flashes  to  see  when  they  are  approaching  coincidence.  As  the 
two  approach  he  notes  the  hours,  minutes,  and  seconds  on  the 
chronometer  when  the  advancing  edge  of  the  moving  flash  touches 
the  first  edge  of  the  fixed  flash.  A  few  seconds  later  he  notes 
when  the  receding  edge  of  the  moving  flash  touches  the  second 
edge  of  the  fixed  flash.  The  mean  of  the  two  gives  the  true  time 
of  coincidence  of  centers  more  accurately  than  it  could  be  ob- 
served directly.  Such  observations  are  made  on  several  succes- 
sive coincidences,  the  flash  moving  alternately  upward  and 
downward.  By  combining  the  up  and  the  down  observations, 
errors  of  adjustment  are  eliminated.  After  a  few  of  these  have 
been  recorded,  the  observer  cuts  out  the  chronometer  and  leaves 
the  pendulum  swinging  for  a  period  of  nearly  eight  hours.  Im- 
mediately after  the  observations  for  coincidences  are  completed, 


CALCULATION  OF  PERIOD  221 

the  temperatures  are  read  on  the  two  thermometers,  and  the 
pressure  is  read  on  the  manometer  tube.  At  the  end  of  the 
eight-hour  period  the  observer  again  observes  a  few  coincidences 
as  well  as  the  arc  (now  diminished  to  about  o°  20'),  the  pressure, 
and  the  temperatures.  It  is  not  necessary  that  he  continue 
observing  throughout  the  whole  eight-hour  period,  because  the 
few  observations  already  referred  to  make  it  possible  to  estimate 
correctly  the  number  of  coincidences  which  must  have  occurred 
between  the  observed  times.  It  is  customary  to  take  the  ob- 
servations with  two  or  more  chronometers  as  a  check. 

This  description  applies  to  the  8-hour  program  outlined  above. 
If  the  pendulums  are  swung  for  a  1 2-hour  period  it  is  necessary 
to  start  each  pendulum  with  a  somewhat  larger  arc  (i°2f)  in 
order  that  it  may  have  a  sufficient  amplitude  at  the  end  of  12 
hours  to  enable  the  observer  to  read  the  coincidences  of  the  flash 
conveniently  and  accurately. 

It  is  desirable  that  the  temperature  of  the  apparatus  be 
kept  as  nearly  uniform  as  possible,  and  that  there  be  little 
vibration.  In  order  to  allow  the  pendulum  time  to  assume  the 
temperature  of  the  receiver  the  next  pendulum  to  be  swung  is 
placed  inside  the  case  before  it  is  used  in  the  observations. 
While  the  case  is  still  in  position  the  observer  must  place  the 
interferometer  in  position  and  observe  the  movement  of  the 
interference  bands  while  the  pendulum  is  swinging. 

149.   Calculation  of  Period. 

After  the  observations  are  complete  and  the  time  observations 
and  the  chronometer  rates  are  computed,  the  time  of  one  oscilla- 
tion for  each  pendulum  in  each  position  is  found  as  follows: 
divide  the  total  number  of  seconds  in  an  Sh  interval  by  the  num- 
ber of  seconds  found  for  one  coincidence  interval  (see  example), 
to  obtain  the  number  of  intervals  that  have  occurred  during  the 
swing.  Since  this  must  be  a  whole  number,  there  will  be  no 
difficulty  in  determining  it  correctly.  Then  reverse  the  process, 
dividing  the  total  interval  by  the  number  of  coincidence  intervals, 
to  obtain  the  accurate  value  of  the  number  of  seconds  (s)  in  one 


222  GRAVITY  MEASUREMENTS 

coincidence  interval.     The  uncorrected  period  of  the  pendulum 
is  found  by 

P  =  —  —  [101] 

2S  —  I 

for  a  sidereal  chronometer,  Table  G,  or 

P  =  —?—  [102] 

2S  +  € 

for  a  mean-time  chronometer. 
150.  Corrections. 

This  period  must  then  be  corrected  to  reduce  it  to  its  value  at 
assumed  standard  conditions,  namely, 
Infinitesimal  arc, 
Temperature  15°  C., 
Pressure  6omm  at  o°  C., 
True  sidereal  time,  and 
Inflexible  support. 

The  correction  to  reduce  P  to  its  value  for  an  infinitesimal  arc 

is 

PM  sin  (<fr  +  <//)  sin  (0  -  <//)       '  ,       , 

32     t  log  sin  4>  —  log  sin  $' 

a  formula  given  by  Borda,  in  which  P  =  the  period,  M  =  the 
modulus  of  the  common  system  of  logarithms,  and  <£  and  0'  = 
the  initial  and  final  arcs. 
The  temperature  correction  is 


T°  being  the  observed  temperature  centigrade  and  a  the  co-   \ 
efficient  to  be  found  by  trial,     (a  =  +0.000008  34). 
The  pressure  correction  is 

-  Pr  1  ,  [105] 

i  +  0.00367  r°j 


K 


in  which        Pr  =  observed  pressure  in  mm, 

T°  =  temperature  centigrade, 
and  K  =  coefficient  to  be  found  by  trial. 


CORRECTIONS  223 

The  constant  0.00367  is  the  coefficient  of  expansion  of  air  for 
i°C. 
The  rate  correction  is  given  by  the  expression 

+  0.000011574  RP,  [106] 

where  R  =  daily  rate  of  chronometer  on  sidereal  time,  +  when 
losing  and  —  when  gaining.  The  coefficient  is  the  reciprocal  of 
the  number  of  seconds  in  one  day. 

The  flexure  correction  is  computed  by  dividing  the  observed 
movement  of  the  fringe  band  (in  scale  divisions)  by  the  width  of 
a  band  and  then  reducing  this  to  an  arc  of  5mm  by  dividing  by  the 
observed  arc  and  multiplying  by  5.  The  result  is  the  displace- 
ment for  a  5rom  arc  in  terms  of  the  width  of  a  band.  This  dis- 
placement, multiplied  by  the  coefficient  (173  mentioned  before), 
gives  the  correction  to  be  subtracted  from  P. 


224 


GRAVITY  MEASUREMENTS 


TABLE  D.  —  REDUCTION  OF  SCALE  READING  IN 
MILLIMETERS  TO  MINUTES  OF  ARC 


Scale. 

i.o  mm. 

2.0  mm. 

3.0  mm. 

4.0  mm. 

5-o  mm. 

mm. 
0.0 

12 

23 

35 

46 

58 

O.I 

13 

24 

36 

48 

59 

O.2 

14 

26 

37 

49 

60 

0-3 

15 

27 

38 

50 

61 

0.4 

16 

28 

39 

63 

°-5 

17 

29 

4i 

52 

64 

0.6 

19 

30 

42 

53 

65 

0.7 

2O 

31 

43 

55 

66 

0.8 

21 

32 

44 

56 

67 

0.9 

22 

34 

45 

57 

68 

TABLE  E.    ARC  CORRECTIONS   (ALWAYS  SUBTRACTIVE) 
FOR  HALF-SECOND  PENDULUMS 
Arc  at  Beginning 


Arc 
at 
end. 

90'. 

85'. 

80'. 

75'. 

70'. 

65'. 

60'. 

55'. 

50'. 

45'- 

40'. 

35'- 

30'. 

25'- 

20'. 

5 

10 

12.0 

II.  0 

10.  0 

9-0 

8.1 

7-3 

6.5 

5.8 

5-0 

4-3 

3.6 

3° 

2.4 

1.9 

1.4 

15 

14-4 

13.3 

12.2 

ii.  i 

IO.O 

9.0 

8.0 

7-2 

6.3 

5.4 

4.6 

3-9 

3-2 

20 
25 

16.9 
10  t 

15-6 
17.8 

14-3 
16.4 

13-0 

ICO 

ii.  8 

TO    *1 

10.7 

9.6 

8.6 

7.6 

6.6 

5-7 
6  Q 

4-9 

4-1 

30 

Ay  *o 

21.7 

20.1 

I8.5 

i-O  .  VJ 

17.0 

AO-  / 

15.6 

14.2 

12.9 

ii.  6 

10.4 

9.2 

u.  y 

8.1 

35 

24.1 

22.4 

20.7 

19.2 

17.6 

16.1 

14.6 

13  2 

ii.  8 

40 

26.5 

24.7 

22.9 

21.2 

19-5 

17.9 

16.3 

14.8 

13.3 

45 

29.0 

27.1 

25.2 

23-4 

21.6 

19.9 

18.2 

50 

31.5 

29.4 

27.4 

25-5 

23.6 

21.8 

20.  o 

55 

34  i 

32.0 

29.8 

27.8 

25.8 

60 

36.7 

34-4 

32.2 

30.0 

27.9 

65 

39-4 

37-0 

34.6 

70 

42.1 

39-6 

37-1 

75 

44-9 

80 

47-7 

85 

90 

In  practice  it  is  convenient  to  combine  Tables  D  and  E  into 
a  single  table  computed  for  such  intervals  that  little  interpola- 
tion is  necessary. 


FORM  OF  RECORD  OF   PENDULUM  OBSERVATIONS        225 


TABLE  F.  —  CORRECTION   FOR  PRESSURE 


Temp. 
C. 

50  mm. 

55  mm. 

oo  mm. 

65  mm. 

70  mm. 

75  mm. 

80  mm. 

85  mm. 

90  mm. 

o 

+  10 

+5 

o 

-5 

—  10 

-15 

—  20 

-25 

-30 

I 

+  10 

+5 

o 

-5 

—  10 

-15 

—  20 

-25 

-^30 

2 

+  10 

+5 

0 

-5 

-  9 

-14 

-19 

-24 

29 

3 

II 

6 

+1 

4 

9 

14 

19 

24 

29 

4 

II 

6 

+1 

4 

9 

14 

19 

24 

29 

5 

II 

6 

+1 

4 

9 

14 

19 

24 

28 

6 

II 

6 

+1 

4 

9 

14 

19 

24 

28 

7 

II 

6 

2 

3 

8 

13 

18 

23 

28 

8 

II 

6 

2 

3 

8 

13 

18 

23 

27 

9 

12 

7 

2 

3 

8 

13 

17 

22 

27 

10 

12 

7 

2 

3 

8 

13 

17 

22 

27 

ii 

12 

7 

2 

3 

7 

12 

17 

21 

26 

12 

12 

7 

2 

2 

7 

12 

17 

21 

26 

13 

12 

1 

3 

2 

7 

12 

17 

21 

26 

14 

12 

8 

3 

2 

7 

II 

16 

21 

26 

IS  ' 

13 

8 

3 

2 

6 

II 

16 

20 

26 

16 

13 

8 

3 

2 

6 

II 

16 

20 

25 

17 

13 

8 

4 

6 

II 

15 

2O 

25 

18 

13 

8 

4 

6 

IO 

IS 

2O 

24 

19 

13 

9 

4 

— 

5 

IO 

IS 

2O 

24 

20 

13 

9 

4 

— 

5 

IO 

IS 

2O 

24 

21 

14 

9 

4 

— 

5 

IO 

14 

19 

24 

22 

14 

9 

4 

— 

5 

10 

14 

19 

23 

23 

14 

9 

5 

0 

5 

9 

14 

19 

23 

24 

14 

9 

5 

O 

4 

9 

14 

18 

23 

25 

14 

10 

5 

o 

4 

9 

13 

18 

22 

26 

14 

10 

5 

+1 

4 

9 

13 

18 

22 

27 

H 

10 

+1 

4 

8 

13 

17 

22 

28 

+  15 

+10 

+6 

+1 

-  4 

-  8 

—  13 

-17 

22 

29 

+  15 

+  10 

+6 

+1 

-  3 

-18 

—  12 

-17 

—  21 

30 

+  15 

+  10 

+6 

+1 

-  3 

-  8 

—  12 

-17 

—  21 

Body  of  table  gives  corrections  (in  yth  decimal  place  of  sec- 
onds) to  period  of  half  seconds  pendulum. 


226 


GRAVITY  MEASUREMENTS 


TABLE  G.  —  PERIODS  OF  QUARTER  METER  PENDULUM 

NOTE  :  To  obtain  period  to  7th  decimal  place,  prefix  .50  or  .500  to  figures  in  the  table. 

Body  of  table  gives 


o 

220O 

2300 

2400 

2500 

2600 

2700 

2800 

2900 

3000 

3100 

o 

11,390 

10,893 

10,438 

10,020 

9634 

9276 

8944 

8636 

8347 

8078 

I 

84 

89 

34 

16 

30 

73 

4i 

33 

44 

75 

2 

79 

84 

30 

12 

26 

70 

38 

30 

42 

72 

3 

74 

79 

25 

08 

23 

66 

35 

27 

39 

70 

4 

69 

74 

21 

04 

19 

63 

32 

24 

36 

67 

5 

11,364 

10,870 

10,417 

10,000 

9615 

9259 

8929 

8621 

8333 

8064 

6 

58 

65 

12 

9996 

12 

56 

25 

18 

30 

62 

7 

53 

60 

08 

92 

08 

52 

22 

15 

28 

59 

8 

48 

55 

04 

88 

04 

49 

19 

12 

25 

57 

9 

43 

5i 

I0>399 

84 

OI 

46 

16 

09 

22 

54 

10 

n,338 

10,846 

io,395 

9980 

9597 

9242 

8913 

8606 

8320 

8052 

ii 

33 

4i 

91 

76 

93 

39 

10 

03 

17 

49 

12 

28 

37 

86 

72 

90 

35 

06 

OO 

14 

46 

13 

22 

32 

82 

68 

86 

32 

03 

8597 

II 

44 

14 

17 

27 

78 

64 

82 

28 

oo 

94 

08 

4i 

is 

11,312 

10,822 

io,373 

9960 

9578 

9225 

8897 

859i 

8306 

8039 

16 

07 

18 

69 

56 

75 

22 

94 

88 

03 

3B 

17 

02 

13 

65 

52 

7i 

18 

9i 

85 

OO 

33 

18 

H,297 

08 

61 

48 

68 

15 

87 

82 

8297 

3i 

i9 

92 

04 

56 

44 

64 

12 

84 

79 

95 

28 

20 

11,287 

io,799 

10,352 

9940 

9560 

9208 

8881 

8576 

8292 

8026 

21 

82 

94 

48 

36 

57 

05 

78 

73 

89 

23 

22 

76 

90 

43 

32 

53 

OI 

75 

70 

86 

20 

23 

72 

85 

39 

28 

49 

9198 

72 

68 

84 

18 

24 

66 

80 

35 

25 

46 

95 

68 

64 

81 

15 

25 

11,261 

10,776 

10,331 

9921 

9542 

9i9i 

8865 

8562 

8278 

8013 

26 

56 

7i 

26 

17 

38 

88 

62 

59 

75 

10 

27 

5i 

67 

22 

13 

35 

84 

59 

56 

73 

08 

28 

46 

62 

18 

09 

3i 

81 

56 

53 

70 

05 

29 

4i 

57 

14 

05 

27 

78 

53 

5o 

67 

03 

30 

11,236 

io,753 

10,309 

9901 

9524 

9174 

8850 

8547 

8264 

8000 

31 

3i 

48 

05 

9897 

20 

7i 

46 

44 

62 

7997 

32 

26 

44 

OI 

93 

17 

68 

43 

4i 

59 

95 

33 

21 

39 

10,297 

89 

13 

64 

40 

38 

56 

92 

34 

16 

34 

92 

85 

09 

61 

37 

35 

53 

90 

35 

11,211 

10,730 

10,288 

9881 

95o6 

9i58 

8834 

8532 

8251 

7987 

FORM  OF   RECORD  OF  PENDULUM  OBSERVATIONS        227 


WHEN   PENDULUM   IS  SLOWER  THAN   CHRONOMETER 

Top  and  left-hand  arguments  combined  give  interval  s  =  ten  coincidence  intervals. 
t  =  period  in  seconds. 


3200 

3300 

3400 

3500 

3600 

3700 

3800 

390° 

4000 

4100 

4200 

0 

7825 

7587 

7364 

7153 

6954 

6766 

6588 

6418 

6258 

6105 

5960 

o 

22 

85 

62 

5i 

52 

64 

86 

17 

56 

04 

58 

I 

2O 

83 

59 

49 

So 

62 

84 

15 

55 

02 

57 

2 

17 

80 

57 

'  47 

48 

60 

82 

14 

53 

OI 

55 

3 

IS 

78 

55 

45 

46 

59 

81 

.  12 

52 

6099 

54 

4 

78l2 

7576 

7353 

7H3 

6944 

6757 

6579 

6410 

6250 

6098 

5952 

5 

IO 

74 

5i 

4i 

42 

55 

77 

09 

48 

96 

Si 

6 

08 

7i 

49 

39 

4i 

53 

76 

07 

47 

95 

5° 

7 

05 

69 

46 

37 

•  39 

5i 

74 

05 

45 

93 

48 

8 

03 

67 

44 

35 

37 

49 

72 

04 

44 

92 

47 

9 

7800 

7564 

7342 

'  7133 

6935 

6748 

6570 

6402 

6242 

6090 

5945 

10 

7798 

62 

40 

3i 

33 

46 

69 

OO 

4i 

89 

44 

ii 

96 

60 

38 

29 

3i 

44 

67 

6399 

39 

87 

42 

12 

93 

58 

36 

27 

29 

42 

65 

97 

38 

86 

4i 

13 

91 

55 

34 

25 

27 

40 

63 

96 

36 

84 

40 

14 

7788 

7553 

7331 

7123 

6925 

6738 

6562 

6394 

6234 

6083 

5938 

15 

86 

5i 

29 

21 

23 

37 

60 

92 

33 

81 

37 

16 

83 

48 

27 

18 

21 

35 

58 

9i 

3i 

80 

35 

i7 

81 

46 

25 

16 

I9 

33 

56 

89 

30 

78 

34 

18 

78 

44 

23 

14 

18 

31 

55 

87 

28 

77 

33 

19 

7776 

7542 

732i 

7112 

6916 

6730 

6553 

6386 

6227 

6075 

593i 

20 

74 

39 

19 

IO 

14 

28 

5i 

84 

25 

74 

30 

21 

7i 

37 

16 

08 

12 

26 

50 

82 

24 

72 

28 

22 

69 

35 

14 

06 

IO 

24 

48 

81 

22 

7i 

27 

23 

66 

32- 

12 

04 

08 

22 

46 

79 

20 

70 

26 

24 

7764 

7530 

7310 

7102 

6906 

6720 

6544 

6378 

6219 

6068 

5924 

25 

62 

28 

08 

oo 

04 

19 

43 

76 

17 

66 

23 

26 

59 

26 

06 

7098 

O2 

17 

4i 

74 

16 

65 

21 

27 

57 

23 

04 

96 

OO 

15 

39 

73 

14 

64 

20 

28 

7754 

752i 

OI 

94 

6898 

13 

38 

7i 

13 

62 

19 

29 

7752 

7519 

7299 

7092 

6897 

6711 

6536 

6369 

6211 

6061 

5917 

30 

So 

16 

97 

90 

95 

IO 

34 

68 

IO 

59 

l6 

31 

47 

14 

95 

88 

93 

08 

32 

66 

08 

58 

!<: 

32 

45 

12 

93 

86 

9i 

06 

3i 

64 

07 

56 

13 

33 

42 

IO 

9i 

84 

89 

04 

29 

63 

05 

55 

12 

34 

7740 

7508 

7289 

7082 

6887 

6702 

6527 

6361 

6204 

6053 

5910 

35 

228 


GRAVITY  MEASUREMENTS 


TABLE  G   (Cow.)-  — PERIODS   OF  QUARTER  METER  PENDU- 

NOTE  :  To  obtain  period  to  7th  decimal  place,  prefix  .50  or  .500  to  figures  in  the  table. 

Body  of  table  gives 


0 

2200 

2300 

2400 

2500 

2600 

2700 

2800 

2900 

3000 

3100 

36 

06 

25 

84 

78 

02 

54 

3i 

30 

48 

85 

37 

OI 

20 

80 

74 

9498 

5i 

28 

27 

45 

82 

38 

11,196 

16 

75 

70 

95 

48 

24 

24 

43 

80 

39 

91 

ii 

7i 

66 

9i 

44 

21 

21 

40 

77 

40 

11,186 

10,707 

10,267 

9862 

9488 

9141 

88l8 

8518 

8237 

7974 

4i 

81 

02 

63 

58 

84 

38 

IS 

15 

34 

72 

42 

76 

10,698 

58 

54 

81 

34 

12 

12 

32 

69 

43 

7i 

93 

54 

50 

77 

3i 

09 

09 

29 

67 

44 

66 

88 

50 

46 

73 

27 

06 

06 

26 

64 

45 

11,161 

10,684 

10,246 

9842 

9470 

9124 

8803 

8503 

8224 

7962 

46 

56 

79 

42 

39 

66 

21 

00 

00 

21 

59 

47 

Si 

75 

38 

35 

62 

18 

8797 

8498 

18 

57 

48 

46 

70 

33 

3i 

59 

14 

94 

95 

16 

54 

49 

4i 

66 

29 

27 

55 

ii 

90 

92 

13 

52 

50 

11,136 

10,661 

10,225 

9823 

9452 

9108 

8787 

8489 

8210 

7949 

5i 

3i 

56 

21 

19 

48 

04 

84 

86 

08 

47 

52 

26 

52 

17 

16 

45 

01 

81 

83 

05 

44 

S3 

21 

47 

12 

12 

4i 

9098 

78 

80 

02 

42 

54 

16 

43 

08 

08 

38 

94 

75 

78 

8i99 

39 

55 

n,  in 

10,638 

IO,2O4 

9804 

9434 

9091 

8772 

8475 

8i97 

7936 

56 

06 

34 

IO,2OO 

9800 

30 

88 

69 

72 

94 

34 

57 

OI 

29 

10,196 

9796 

27 

84 

66 

69 

9i 

32 

58 

11,096 

25 

92 

92 

23 

81 

63 

66 

89 

29 

59 

91 

20 

88 

88 

20 

78 

60 

63 

86 

26 

60 

I  I,  086 

10,616 

10,183 

9785 

9416 

9074 

8757 

8460 

8183 

7924 

61 

82 

ii 

79 

81 

13 

7i 

54 

57 

81 

21 

62 

77 

07 

75 

77 

09 

68 

5i 

54 

78 

19 

63 

72 

02 

7i 

73 

06 

65 

47 

52 

75 

16 

64 

67 

10,598 

67 

69 

O2 

61 

44 

49 

73 

14 

65 

11,062 

IQ.593 

10,163 

9766 

9398 

9058 

8741 

8446 

8170 

79ii 

66 

57 

89 

59 

62 

95 

55 

38 

43 

67 

09 

67 

52 

84 

54 

58 

92 

5i 

35 

40 

65 

06 

68 

47 

80 

50 

54 

88 

48 

32 

37 

62 

04 

69 

42 

75 

46 

So 

84 

45 

29 

34 

59 

01 

70 

11,038 

10,571 

10,142 

9747 

938i 

9042 

8726 

8432 

8i57 

7899 

FORM  OF   RECORD  OF   PENDULUM  OBSERVATIONS        229 


LUM  WHEN   PENDULUM   IS   SLOWER  THAN  CHRONOMETER 

Top  and  left-hand  arguments  combined  give  interval  s  =  ten  coincidence  intervals. 
/  =  period  in  seconds. 


3200 

3300 

3400 

3500 

3600 

3700 

3800 

3900 

4000 

4100 

4200 

o 

38 

05 

86 

80 

85 

OI 

26 

60 

O2 

52 

09 

36 

35 

03 

84 

78 

83 

6699 

24 

58 

00 

5o 

07 

37 

33 

OI 

82 

76 

8! 

.  97 

22 

56 

6199 

49 

06 

38 

3° 

7498 

80 

74 

80 

95 

21 

55 

97 

47 

05 

39 

7728 

7496 

7278 

7072 

6878 

6693 

6519 

6353 

6196 

6046 

5903 

40 

26 

94 

76 

70 

76 

92 

17 

52 

94 

44 

O2 

4i 

23 

92 

74 

68 

74 

90 

16 

5o 

93 

43 

oo  42 

21 

9<> 

72 

66 

72 

88 

14 

48 

9i 

42 

5899  43 

18 

87 

70 

64 

70 

86 

12 

47 

90 

40 

98 

44 

7716 

7485 

7267 

7062 

6868 

6684 

6510 

6345 

6188 

6039 

5896 

45 

14 

83 

65 

60 

66 

83 

09 

44 

87 

37 

95 

46 

ii 

80 

63 

58 

64 

81 

07 

42 

85 

36 

93 

47 

09 

78 

61 

56 

62 

79 

05 

40 

84 

34 

92 

48 

06 

76 

59 

54 

61 

77 

04 

39 

82 

33 

9i 

49 

7704 

7474 

7257 

7052 

6859 

6676 

6502 

6337 

6180 

6031 

5889 

50 

02 

72 

55 

50 

57 

74 

00 

36 

79 

30 

88 

5i 

7699 

69 

53 

48 

55 

72 

6499 

34 

77 

28 

86 

52 

97 

67 

Si 

46 

53 

70 

97 

32 

76 

27 

85 

53 

95 

65 

48 

44 

51 

69 

95 

3i 

74 

26 

84 

54 

7692 

7463 

7246 

7042 

6849 

6667 

6494 

6329 

6173 

6024 

5882 

55 

90 

60 

44 

40 

47 

65 

92 

28 

7i 

23 

81 

56 

88 

58 

42 

38 

46 

63 

90 

26 

70 

21 

80 

57 

85 

56 

40 

36 

44 

61 

88 

24 

68 

2O 

78 

58 

83 

7680 

54 
7452 

38 

7236 

34 
7032 

42 
6840 

60 
6658 

87 
6485 

,  23 
6321 

67 
6165 

18 
6017 

77 
5875 

g 

78 

49 

34 

30 

38 

56 

83 

20 

64 

IS 

74 

61 

76 

47 

32 

28 

36 

54 

82 

18 

62 

14 

73 

62 

73 

45 

30 

26 

34 

52 

80 

1  6 

61 

12 

7i 

63 

7i 

43 

28 

24 

32 

51 

78 

15 

59 

II 

70 

64 

7669 

7440 

7225 

7022 

6831 

6649 

6477 

6313 

6158 

6010 

5868 

65 

66 

38 

23 

20 

29 

47 

75 

12 

56 

08 

67 

66 

64 

36 

21 

18 

27 

45 

73 

IO 

55 

07 

66 

67 

62 

34 

19 

17 

25 

44 

72 

08 

53 

05 

64 

68 

59 

32 

17 

15 

23 

42 

70 

07 

52 

04 

63 

69 

7657 

7429 

7215 

7013 

6821 

6640 

6468 

6305 

6150 

6OO2 

5862 

70 

230 


GRAVITY  MEASUREMENTS 


TABLE  G   (Cow.).  — PERIODS  OF  QUARTER  METER  PENDU- 

NOTE  :  To  obtain  period  to  7th  decimal  place,  prefix  .50  or  .500  to  figures  in  the  table. 

Body  of  table  gives 


o 

2200 

2300 

2400 

2500 

2600 

2700 

2800 

2900 

3000 

3100 

71 

33 

66 

38 

43 

77 

38 

23 

29 

54 

96 

72 

28 

62 

34 

39 

74 

35 

20 

26 

5i 

94 

73 

23 

57 

30 

35 

70 

32 

17 

23 

49 

9i 

74 

18 

53 

26 

3i 

67 

29 

14 

20 

46 

89 

75 

11,013 

10,548 

10,122 

9728 

9363 

9025 

8711 

8418 

8i43 

7886 

76 

08 

44 

17 

24 

60 

22 

08 

15 

4i 

84 

77 

04 

40 

13 

20 

56 

19 

05 

12 

38 

81 

78 

10,999 

35 

09 

16 

53 

16 

02 

09 

35 

79 

79 

94 

3i 

05 

12 

49 

12 

8699 

06 

33 

76 

80 

10,989 

10,526 

IO,IOI 

9709 

9346 

9009 

8696 

8403 

8130 

7874 

81 

84 

22 

10,097 

05 

42 

06 

93 

OI 

28 

72 

82 

79 

18 

93 

OI 

39 

02 

90 

8398 

25 

69 

83 

74 

13 

89 

9697 

35 

8999 

87 

95 

22 

67 

84 

70 

09 

85 

94 

32 

96 

84 

92 

2O 

64 

85 

10,965 

10,504 

10,081 

9690 

9328 

8993 

8681 

8389 

8lI7 

7862 

86 

60 

10,500 

77 

86 

25 

90 

78 

86 

14 

59 

87 

55 

10,495 

73 

82 

21 

86 

75 

84 

12 

57 

88 

5i 

9i 

68 

79 

18 

83 

72 

81 

09 

54 

89 

46 

87 

64 

75 

14 

80 

69 

78 

06 

52 

90 

10,941 

10,482 

10,060 

9671 

93" 

8977 

8665 

8375 

8104 

7849 

9i 

36 

78 

56 

68 

08 

74 

62 

72 

OI 

47 

92 

3i 

73 

52 

64 

04 

70 

60 

70 

8098 

44 

93 

27 

69 

48 

60 

OI 

67 

56 

67 

96 

42 

94 

22 

65 

44 

56 

9297 

64 

54 

64 

93 

40 

95 

10,917 

10,460 

10,040 

9653 

9294 

8961 

8650 

8361 

8091 

7837 

96 

12 

56 

36 

49 

90 

57 

48 

58 

88 

34 

97 

08 

52 

32 

45 

87 

54 

44 

56 

85 

32 

98 

03 

47 

28 

4i 

83 

Si 

42 

53 

83 

30 

99 

10,898 

43 

24 

38 

80 

48 

39 

50 

80 

27 

100 

10,893 

10,438 

10,020 

9634 

9276 

8944 

8636 

8347 

8078 

7825 

FORM   OF  RECORD  OF   PENDULUM  OBSERVATIONS        231 


LUM  WHEN    PENDULUM   IS  SLOWER  THAN   CHRONOMETER 

Top  and  left-hand  arguments  combined  give  interval  s  =  ten  coincidence  intervals. 
t  =•  period  in  seconds. 


3200 

33oo 

3400 

3500 

360O 

3700 

3800 

3900 

4000 

4100 

4200 

O 

55 

27 

13 

II 

19 

38 

67 

04 

49 

OI 

60 

71 

52 

25 

ii 

09 

18 

37 

65 

02 

47 

oo 

59 

72 

5o 

23 

09 

07 

16 

35 

63 

oo 

46 

5998 

58 

73 

48 

21 

07 

05 

14 

33 

61 

6299 

44 

97 

56 

74 

7645 

7418 

7205 

7003 

6812 

6631 

6460 

6297 

6142 

5995 

5855 

75 

43 

16 

02 

OI 

10 

3° 

58 

96 

4i 

94 

53 

76 

4i 

14 

oo 

6999 

08 

28 

57 

94 

40 

92 

52 

77 

38 

12 

7198 

97 

06 

26 

55 

£2 

38 

9i 

5i 

78 

36 

10 

96 

95 

05 

24 

53 

9i 

36 

89 

49 

79 

7634 

7407 

7194 

6993 

6803 

6622 

6452 

6289 

6i35 

5988 

5848 

80 

3i 

05 

92 

9i 

OI 

21 

So 

88 

34 

87 

47 

81 

29 

03 

90 

89 

6799 

19 

48 

86 

32 

85 

45 

82 

27 

01 

88 

87 

97 

17 

47 

85 

3° 

84 

44 

83 

24 

7399 

86 

85 

95 

16 

45 

83 

29 

.  82 

42 

84 

7622 

7396 

7184 

6983 

6794 

6614 

6443 

6281 

6128 

598i 

5841 

85 

20 

94 

82 

81 

92 

12 

42 

80 

26 

80 

40 

86 

17 

92 

80 

79 

IO 

40. 

78 

24 

78 

38 

87 

IS 

90 

78 

77 

88 

09 

38 

77 

23 

77 

37 

88 

13 

88 

76 

75 

86 

07 

37 

75 

22 

75 

36 

89 

7610 

7386 

7174 

6974 

6784 

6605 

6435 

6274 

6l20 

5974 

5834 

90 

08 

83 

72 

72 

82 

03 

33 

72 

18 

72 

33 

9i 

06 

81 

70 

70 

81 

O2 

32 

70 

17 

7i 

32 

92 

03 

79 

67 

68 

79 

6600 

30 

69 

16 

69 

3° 

93 

OI 

77 

65 

66 

77 

6598 

28 

67 

14 

68 

29 

94 

7599 

7375 

7163 

6964 

6775 

6596 

6427 

6266 

6112 

5967 

5828 

95 

96 

72 

61 

62 

73 

95 

25 

64 

ii 

65 

26 

96 

94 

70 

59 

60 

7i 

93 

23 

62 

IO 

64 

25 

97 

92 

68 

57 

58 

70 

9i 

22 

61 

08 

62 

23 

98 

90 

66 

55 

56 

68 

89 

20 

59 

06 

61 

22 

99 

7587 

7364 

7153 

6954 

6766 

6588 

6418 

6258 

6105 

5960 

5821 

IOO 

232 


GRAVITY  MEASUREMENTS 


151.  Form  of  Record  cf  Pendulum  Observations. 

Following  is  a  specimen  record  of  a  single  swing  made  with 
''Apparatus  B,"  belonging  to  the  Coast  Survey. 

Station:  Sawah  Loento,  Sumatra.     Date:  May  7,  1901. 
Observer:  G.  L.  H.     Chronometer:  Bond  541  (sid.) 

Pendulum  B  4,  Direct,  on  Knife  edge  / 


Observed  coincidences. 

Pressure. 

Temperature. 

Arc. 

h      m    s 

mm. 

(C). 

mm. 

D    9  59  03 

U   10   02    12 

27-5 

D        05  ii 

27.5 

22°  .  6 

4-5  =  52' 

U        08  18 

55-o 

D            II    12 

U        14  19 

D    4  54  42 

U        58  12 

28.0 

D    5  oo  43 

28.0 

28.8 

0.9  =  10' 

U       04  08 

56.0 

D       06  42 

U        10  06 

55-5 
4-2 
51.3  ato°C. 


Ther.  error 


25.70 
-•30 


Total  interval  (mean)  6^  55™  43*  =  24,943*. 

Approximate  length  of  coincidence  interval  =  3"*  01*  =  181*. 

Number  of  coincidence  intervals  =  138. 

Length  of  one  coincidence  interval  =  180.75. 

Period  (uncorrected)  =  0.5013869. 


Uncorrected  Period 
Corr.  for  Arc 

"      "  Temp. 

"      "  Press. 

"      "   Rate  (No.  541) 

"      "   Flexure 
Corrected  Period  = 


REDUCTION  TO  SEA-LEVEL  233 

152.  Calculation  of  g. 

After  the  period  has  been  corrected  for  instrumental  errors, 
the  value  of  gravity  (g)  may  be  found  by  comparing  the  period 
(P)  with  that  of  the  same  pendulum  at  some  point  where  the 
value  of  g  is  known,  say  at  Washington.  If  the  value  at  Wash- 
ington is  gwj  then 

E>2 
•*•     w  r  i 

S=--Jf&-  I10?] 

Evidently  it  is  of  the  greatest  importance  that  the  period  should 
not  change  during  a  series  of  observations  made  for  the  purpose 
of  comparing  P  at  different  stations.  The  pendulum  should  be 
swung  at  frequent  intervals  at  the  base  station,  to  test  its  in- 
variability; in  any  case  it  should  be  swung  at  the  beginning  and 
end  of  every  series. 

Example.  Suppose  that  the  mean  corrected  period  of  a  set  of  pendulums  at  a 
station  is  0.5012480,  and  at  Washington,  the  base  station,  is  0.5007248,  and  that  gw 
is  taken  as  980.111  dynes.  Then,  by  formula  [107],  g  =  978.066  dynes. 

153.  Reduction  to  Sea-Level. 

The  value  of  gravity  found  in  the  manner  just  described  is  the 
value  at  the  station,  assuming  the  length  of  the  pendulum  to  be 
invariable  and  the  chronometer  correction  to  be  correct.  In 
comparing  values  at  different  stations,  however,  it  is  essential  to 
reduce  the  observed  value  to  the  value  at  sea-level.  A  formula 
long  used  for  this  purpose  is  one  devised  by  Bouguer  when  re- 
ducing observations  made  along  the  Peruvian  arc  in  1749.  This 
formula  is 


in  which  H  is  the  elevation  of  the  station  above  sea-level, 

r  is  the  radius  of  the  earth, 

5  is  the  density  at  the  surface, 
and  A  is  the  mean  density  of  the  earth. 

The  first  term  of  this  formula  allows  for  the  decrease  in  gravity 
due  to  height  alone;  the  second  term,  for  the  increase  in  attraction 
due  to  the  topography  beneath  the  station. 


234  GRAVITY  MEASUREMENTS 

The  correction  for  height  of  station  is  derived  from  the  law  of 
gravitation,  namely  that  the  force  of  attraction  varies  inversely 
as  the  square  of  the  distance;  whence 


Therefore  go  =  g  (i  +  ^)  -  [109] 

The  correction  for  topography  is  based  upon  the  assumption 
that  it  is  due  to  the  attraction  of  a  cylinder  whose  axis  is  vertical 
and  whose  height  is  small  compared  with  its  width.  The  at- 
traction on  a  unit  mass  at  the  station  is  shown  by  Helmert  (Hohe. 
GeocasietV(A.  II,  pp.  142  and  164)  to  be 

Ag  =  2  irkdH.  (a) 

The  attraction  of  the  sphere  on  the  same  mass  is 


.  (b) 

r  3 


Dividing  (a)  by  (b)  and  multiplying  by  g, 


Adding  both  corrections  ([109]  and  [no])  and  remembering  that 
the  two  are  of  opposite  sign, 

2H         3    5    H 

£n  =   a  4-  a  --   a  £  .  —  .  — 

r^    r        g2   A    r 


Another  method  of  reduction  which  has  been  much  used  is  to 
omit  the  last  term  of  Bouguer's  formula,  and  correcting  for 
height  only.  In  this  case  the  correction  to  g  is 

^  .     2  H  r          , 

Corr.  =  H  --  g,  [112] 

or  Corr.  =  +0.0003086  H  (meters).  [1120] 

*  See  also  Clarke,  Geodesy,  p.  325.  For  an  additional  term  for  irregularity  in 
topography  see  Coast  Survey  Report  for  1894,  p.  22. 


CALCULATION  OF  THE  COMPRESSION  235 

This  method  was  introduced  because  the  former  method 
showed  large  disagreement  between  observed  and  computed 
values.  The  second,  or  ''free-air,"  method  showed  better  agree- 
ments, indicating  a  compensation  due  to  variations  of  density 
beneath. 

The  method  employed  by  Professor  Hayford  in  the  Coast 
Survey  investigation  shows  that  still  better  agreement  is  obtained 
by  the  introduction  of  the  assumption  of  isostasy.  The  results 
corrected  by  this  method  show  a  close  general  agreement,  but  in 
certain  localities  there  is  evidence  that  the  isostatic  adjustment 
is  imperfect  —  for  example,  near  Seattle  in  the  United  States 
and  at  certain  places  near  the  Himalayas  in  India. 

154.   Calculation  of  the  Compression. 

By  employing  a  large  number  of  observed  values  of  g  the  most 
probable  values  of  the  constants  ge  and  gp  may  be  found.  From 
these  data  the  compression  may  be  derived  by  applying  Clairaut's 
formula, 


2      fa  P 

z      6«  6 


/2TT\2 

The  value  of  ce  is  f  —  1  •  a,  where  T  =  86164.09  seconds  and  a  is 

the  equatorial  radius.     Using  Clarke's  value  of  a,  the  resulting 
value  of  ce  is  found  to  be 

ce  =  0.033916, 
ana  using  for  ge  the  value  978.038,*  we  obtain 


Then  for  the  compression,  we  have 

a  —  b         i 


a          297.1 

If  the  more  accurate  form  [980]  of  Clairaut's  equation  is  em- 
ployed, the  result  is 

a  —  b  _      i 

a          298.2 
*  See  Coast  Survey  Special  Publication  No.  12. 


236 


GRAVITY  MEASUREMENTS 


By  studying  a  large  number  of  gravity  observations  in  all  parts 
of  the  world  Helmert  obtained  the  value 

a  —  b i 

a  298.3  ±  0.7 

In  the  publication  entitled  Effect  of  Topography  and  Isostatic 
Compensation  upon  the  Intensity  of  Gravity  the  authors  give 

a  —  b i 

a          298.4  ±1.5 

In  the  most  recent  report  on  gravity  work  (Coast  Survey 
Special  Publication  No.  40,  1917),  the  compression  calculated 
from  the  observations  in  the  United  States,  Canada,  Europe  and 

India  is 

a  —  b  I  r       -, 

a       ~  297.4 

By  employing  Equa.  [88]  the  value  of  g  may  be  computed  for 
each  station  on  the  assumption  that  the  earth  is  a  spheroid.  A 
comparison  at  each  station  of  the  observed  and  computed  values 
of  gravity  indicates  to  what  extent  the  geoid  departs  from  the 
spheroid  at  each  point. 


Problem  i. 


PROBLEMS 

Compute  —    —  from  the  following  data: 


Station. 

go- 

Latitude. 

Umanak,  Greenland 

082    «?Qi? 

+  7O   4O    2Q 

Sawah  Loento,  Sumatra 

Q?8   O57 

—  oo  41   40 

Problem  2.     If  the  coincidence  intervals  are  5™  during  an  8-hour  swing,  what  will 
be  the  error  ih  P  due  to  an  error  of  i8  in  noting  the  time  of  a  coincidence? 


CHAPTER  X 
PRECISE  LEVELING  — TRIGONOMETRIC  LEVELING 

155.   Precise  Leveling. 

The  term  precise  leveling  is  applied  to  the  operation  of  deter- 
mining differences  in  elevation  of  successive  points  on  the  earth's 
surface  with  instruments  and  methods  which,  though  similar  to 
those  used  in  ordinary  leveling,  are  more  refined  and  capable  of 
yielding  a  much  higher  degree  of  precision.  In  order  to  secure 
the  greatest  possible  accuracy,  it  is  necessary  to  modify  our  con- 
ception of  the  nature  of  a  level  surface  and  to  introduce  certain 
corrections  which  are  ordinarily  negligible.  It  should  be  ob- 
served that  since  the  line  of  sight  of  the  instrument  is  always 
theoretically  perpendicular  to  the  direction  of  gravity  at  each 
station,  it  lies  in  a  plane  which  is  tangent  to  the  geoid,  not  to  the 
spheroid.  In  tracing  out  a  level  line  by  means  of  the  spirit  level 
we  are  following  the  curvature  of  the  geoidal  surface. 

The  term  precise  leveling  has  for  many  years  been  applied  to 
all  leveling  of  a  fairly  high  degree  of  precision,  but  there  have 
been  various  limits  of  precision  prescribed  by  the  different  or- 
ganizations carrying  on  the  work.  The  accuracy  obtainable  has 
been  so  greatly  increased  through  recent  developments  in  instru- 
ments and  methods  that  in  1912  a  new  class  of  leveling,  known  as 
leveling  of  high  precision,  was  established  by  the  International 
Geodetic  Association;  it  is  to  include  every  line,  set  of  lines,  or 
net,  which  is  run  twice  in  opposite  directions,  on  different  dates, 
and  whose  errors,  both  accidental  and  systematic,  computed  in 
accordance  with  formulas  stated  in  the  resolution,*  do  not  exceed 
dbimm  per  kilometer  for  the  probable  accidental  error  and  dbo.2mm 
per  kilometer  for  the  probable  systematic  error. 

*  See  Coast  Survey  Special  Publication  No.  18,  p.  88.  See  also  Report  of  In- 
ternational Geodetic  Association  for  1912. 

237 


238        PRECISE  LEVELING — TRIGONOMETRIC  LEVELING 

Many  different  instruments  have  been  used  in  the  past  for 
precise  leveling,  some  of  the  "wye  "  type  and  some  of  the 
"dumpy "  type.  All  precise  levels,  however,  have  certain 


characteristics  in  common:  namely,  (i)  a  telescope  of  high  mag- 
nifying power,  mounted  on  a  heavy  tripod:  (2)  a  sensitive  spirit 
level;  (3)  a  slow-motion  screw  for  centering  the  bubble;  (4)  stadia 


PRECISE  LEVELING 


239 


wires  for  determining  the  length  of  sight;  and  (5)  a  mirror  or  other 
optical  device  for  viewing  the  bubble  from  the  eye  end  of  the 
telescope.  Before  the  year  1 899  the  precise  leveling  of  the  United 
States  Coast  Survey  was  done  with  a  wye  level  and  target  rods. 
The  target  was  not  set  exactly  on  the  level  of  the  instrument,  but 


FIG.  Spa.    Precise  Level. 
(C.  L.  Berger  and  Sons.) 

was  set  approximately,  and  corrections  to  this  approximate  read- 
ing were  determined,  using  the  micrometer  screw  to  measure  the 
small  vertical  angles.  Since  1899  *  a  dumpy  level  of  new  design 
has  been  substituted  for  the  wye  level,  the  self-reading  rod 

*  For  a  discussion  of  this  change  in  methods  see  Coast  Survey  Report  for  1899, 
p.  8,  and  for  a  description  of  the  new  instrument  see  Coast  Survey  Report  for  1900, 
p.  521,  and  for  1903,  p.  200. 


240        PRECISE  LEVELING  — TRIGONOMETRIC  LEVELING 


-a 


adopted,  and  the  micrometer  screw  used  only  for 
centering  the  bubble.  This  new  instrument  and 
method  have  been  adopted  by  several  other 
branches  of  the  government  service. 

156.  Instrument. 

The  new  instrument,  sometimes  called  the  prism 
level,  is  designed  to  reduce,  so  far  as  possible,  any 
errors  arising  from  unequal  heating  of  the  different 
portions  of  the  instrument.  (Fig.  89.)  The  tele- 
scope barrel  is  made  of  an  alloy  of  iron  and  nickel 
having  a  low  coefficient  of  expansion  (0.000004  Per 
i°  C.).  The  level  vial  is  set  into  the  telescope  tube 
as  low  as  possible  without  interfering  with  the  cone 
of  rays  from  the  object  glass.  This  diminishes  the 
effect  of  differential  expansion  of  the  parts  support- 
ing  the  level.  At  one  side  of  the  telescope  is 
another  (similar)  tube  containing  a  pair  of  prisms 
which,  together  with  a  mirror  mounted  above  the 
telescope,  enable  the  observer  to  view  the  ends  of 
the  bubble  with  the  left  eye  at  the  same  time  that 
he  looks  at  the  rod  with  the  right  eye.  The  arrange- 
ment  of  mirror  and  prisms  is  such  that  there  is  no 
parallax  caused  by  the  glass  in  the  level  or  the 
mirror.  The  instrument  is  provided  with  the  usual 
small  levels  for  the  approximate  leveling  of  the 
base. 

157.  Rods. 

The  rods  used  are  of  the  non-extensible  type, 
graduated  to  centimeters  and  marked  so  that  they 
may  be  read  directly  by  the  observer  through  the 
telescope,  the  millimeters  being  estimated.  (Fig. 
90.)  The  rods  are  in  the  form  of  a  cross  (in  sec- 
tion) ;  they  are  treated  with  paraffin  to  make  them 
proof  against  moisture.  Metal  plugs  are  inserted 
three  meters  apart  for  verifying  the  length  of  the 


ADJUSTMENTS  241 

rod.  Each  rod  has  a  spirit  level  attached,  to  show  when  it 
is  vertical,  and  also  a  thermometer,  which  is  read  at  each 
sight. 

158.  Turning  Points. 

Foot-pins  are  carried  by  all  leveling  parties,  to  be  used  when 
other  turning  points  are  not  available.  These  are  about  one 
foot  long,  with  a  depression  at  the  top  in  which  to  hold  the 
rod.  A  rope  run  through  a  small  hole  is  provided  for  pulling 
up  the  pin.  Most  of  the  leveling  of  the  Coast  Survey  is  carried 
along  railroad  lines,  and  the  top  of  a  rail  is  the  usual  turning 
point. 

159.  Adjustments. 

The  adjustments  of  the  level  are  nearly  the  same  as  those  of 
the  ordinary  dumpy  level.  The  rough  levels  are  adjusted  so  as 
to  remain  in  the  center  when  the  telescope  is  revolved  about  the 
vertical  axis.  The  axis  of  the  long  bubble  tube  is  adjusted 
parallel  to  the  line  of  sight  of  the  telescope  whenever  it  is  much 
in  error.  This  adjustment  is  tested  each  day  by  taking  four 
readings,  like  those  used  in  the  "peg  "  method,  except  that  the 
shorter  sights  are  10  meters  in  length  and  the  longer  sights  are  of 
the  usual  length,  (say  ioom).  From  these  four  readings  a  factor 
C  is  computed,  which  is  the  ratio  of  the  correction  for  any  reading 
to  the  corresponding  rod  interval.  The  difference  in  the  sums  of 
the  foresight  and  backsight  at  any  set-up  is  to  be  multiplied  by 
this  factor  C. 

To  find  an  expression  for  C,  call  %  and  ih  the  rod  readings  for 
the  nearer  sights,  and  di  and  d%  the  rod  readings  for  the  distant 
sights,  Si  and  s2  the  nearer  stadia  intervals,  and  Si  and  52  the  dis- 
tant stadia  intervals,  the  subscripts  referring  to  the  first  and 
second  instrument  positions.  Then  the  true  difference  in  eleva- 
tion from  the  first  set-up  is 

(ni  +  Csi)  -  (4  +  C5i), 
and  for  the  second  set-up, 

(4  +  C&)  -  («2  +  Csz). 


242        PRECISE  LEVELING  — TRIGONOMETRIC  LEVELING 


Equating  and  solving  for  C, 
/i      Oh  *t 


-  (di  +  4) 


(•&  +  &)-(*  +  *) 

C  is  -f-  if  the  line  of  sight  is  inclined  downward. 

Below  is  table  showing  a  determination  of  C  (from  Coast  Survey 
Report  for  1903). 

DETERMINATION   OP   C.     8.20A.M.,  AUGUST   28,   1900 


(Left-hand  page.)' 

(Right-hand  page.) 

Number 
of 
station. 

Thread 
reading, 
backsight. 

Mean. 

Thread 
interval. 

Rod. 

Thread 
reading, 
foresight. 

Mean. 

Thread 
interval. 

A 
B 

Corr.  fo 

1515 
1528 
1542 

2252 

2357 
2462 

r  curv.  and  r 

1528.3 

2357-0 
0461  .  7 

13 
14 

27 

105 
105 
2IO 

419 
52 

W 
W 

0357 
0462 
0566 

1276 
1288 
1301 

36; 

0461  .7 

1288.3 
1528.3 

105 
104 
209 

12 
13 

25 

2818.7 
ef.  -0.8 

2816.6 
2817.9 

367 

2817.9 

0-1-3  (—  0.004=  C 

If  the  value  of  C  is  less  than  0.005,  the  instrument  should  not 
be  adjusted.  If  between  0.005  and  o.oio,  the  observer  is  advised 
not  to  adjust.  If  over  o.oio,  the  adjustment  should  be  made. 
The  adjustment  is  made  by  moving  the  level  rather  than  the 
cross-hair  ring,  to  avoid  moving  the  line  of  sight  from  the  optical 
axis. 

160.   Method  of  Observing.* 

It  is  customary  to  use  two  rods,  the  one  that  is  held  for  a  fore- 
sight on  a  certain  turning  point  being  kept  at  the  same  turning 
point  for  a  back  sight.  The  instrument  is  set  up  and  leveled, 

*  The  General  Instructions  for  Precise  Leveling  will  be  found  in  Coast  Survey 
Special  Publication  No.  22,  p.  29. 


COMPUTING  THE   RESULTS  243 

and  all  three  hairs  are  read  on  the  back  rod,  the  level  being  kept 
central  at  each  reading.  As  soon  as  possible  thereafter  the  three 
hairs  are  read  in  a  similar  manner  on  the  forward  rod.  The 
readings  are  estimated  to  miUimeters.  The  temperature  on  the 
rod  thermometer  is  read  at  the  same  time.  The  level  should  be 
shaded  from  the  sun  in  order  to  avoid  unequal  heating  of  its  parts. 
In  selecting  instrument  and  rod  points,  the  observer  must  keep 
the  difference  in  length  of  the  forward  and  backward  sight  less 
than  10  meters  on  any  one  set-up  and  less  than  20  meters  for  the 
accumulated  difference  at  any  time.  The  readings  of  the  upper 
and  lower  (stadia)  wires  enable  the  recorder  to  determine  the 
difference  in  distance  at  each  set-up.  The  maximum  length  of 
sight  allowable  is  150™,  a  distance  reached  only  under  exception- 
ally favorable  conditions.  At  odd-numbered  stations  the  back 
sight  is  taken  first;  at  even-numbered  stations  the  fore  sight  is 
taken  first.  This  results  in  the  same  rod  being  read  first  each 
time. 

Lines  between  bench  marks  are  divided  into  sections  of  from 
one  to  two  kilometers  each.  Each  of  these  sections  is  run  for- 
ward and  backward.  If  the  two  differences  in  elevation  so  de- 
termined are  found  to  differ  by  more  than  4mm  VK  (K  =  kilo- 
meters), both  runnings  must  be  repeated  until  such  a  check  is 
obtained.  Lines  may  be  run  with  such  care  that  it  is  seldom 
necessary  to  repeat,  but  the  maximum  economy  appears  to  be 
reached  when  from  5  to  15  per  cent  of  the  sections  have  to  be  re- 
run. 

On  page  244  is  a  set  of  notes  used  in  leveling  with  this  in- 
strument (Coast  Survey  Report,  1903). 

The  most  recent  practice  is  to  record  the  readings  directly  on 
adding  machines  carried  with  the  leveling  outfit.  This  results 
in  a  saving  of  time  and  in  avoiding  many  mistakes  in  recording 
and  adding. 

161.   Computing  the  Results. 

In  computing  the  results  of  precise  leveling,  corrections  are 
applied  for  the  nonadjustment  of  the  level,  for  curvature  and 


244        PRECISE  LEVELING— TRIGONOMETRIC  LEVELING 


refraction,  for  error  in  length  of  rod,  for  error  due  to  temperature 
of  rod,  and  for  the  orthometric  correction.  The  curvature  and 
refraction  corrections  are  usually  taken  from  tables  (Coast  Survey 
Report,  1903).  The  length  of  rod  is  tested  at  the  office  at  the  be- 
ginning and  end  of  the  season,  and  variations  during  the  season 
are  tested  in  the  field  by  means  of  a  steel  tape.  The  temperature 
correction  is  derived  from  tables,  the  argument  being  the  ob- 
served temperatures. 

SPIRIT   LEVELING 


(Left-hand  page.) 

(Right-hand  page.) 

Date:    August  29,  1900. 

From  B.M.  :  68.              To  B.M.  :  O 

Sun  :  C.       Forward.       Backward: 

Wind  :  S.T. 

(Strike  out  one  word.) 

Thread 

Thread 

XT-       -{ 

read- 

Thread 

Sum  of 

Rod 

read- 

Thread 

Sum  of 

JNO.  OI 

station. 

ing, 

Mean.     [ 

inter- 

inter- 

and 

ing 

Mean. 

inter- 

inter- 

back- 

val. 

vals. 

temp. 

fore- 

val. 

vals. 

sight. 

sight. 

43 

0674 

99 

V 

2683 

99 

0773 

0773.0 

99 

38 

2782 

2782.3 

100 

0872 

198 

2882 

199 

0925 

106 

w 

2415 

103 

44 

1031 

1030.3 

104 

35 

2518 

2518.0 

103 

1135 

210 

408 

2621 

206 

405 

0484 

98 

V 

2510 

96 

45 

0582 

0582.3 

99 

35 

2606 

2606.0 

96 

0681 

197 

605 

2702 

192 

597 

0398 

97 

W 

2859 

96 

46 

0495 

0495.0 

97 

34 

2955 

2954-7 

95 

0592 

194 

799 

3050 

191 

788 

1627 

26 

V 

1006 

29 

47 

1053 

1053-3 

27 

34 

1035 

1034-7 

28 

1080 

53 

852 

1063 

57 

845 

11895.7 

3933-9 

-7961.8 

2  :  25  P.M. 

SOURCES  OF  ERROR 


245 


162.  Bench  Marks. 

The  bench  marks  used  in  precise  leveling  are  of  various  types. 
Wherever  it  is  practicable,  the  metallic  plates  shown  in  Fig.  91 
are  used  to  mark  the  points,  but  nearly  all  of  the  kinds  of  bench 
marks  which  are  used  by  engineers  are  used  also  in  this  class  of 
work.  The  distance  between  benches  is  not  allowed  to  exceed 
15  kilometers;  every  100  kilometer  section  should -have  at  least 
20  bench  marks,  a  good  average  distance  being  2.5  kilometers. 
In  cities  the  old  bench  marks  are  often  utilized  for  the  precise 
levels. 

163.  Sources  of  Error. 

The  sources  of  error  which  it  is  particularly  necessary  to  study 
in  this  class  of  work  are  (i)  unequal  effects  of  temperature  changes 
in  the  instrument,  (2)  gradual  rising  or  settling  of  the  instrument 
or  rods,  (3)  variations  in  refraction  of  the  air,  (4)  unequal  lengths 
of  sights,  (5)  errors  in  length  and  temperature  of  rod,  and  (6) 
convergence  of  level  surfaces. 


TABLE  H.  —  TOTAL  CORRECTION  FOR  CURVATURE 
AND  REFRACTION 


Distance. 

Correction  to  rod 
reading. 

Distance. 

Correction  to  rod 
reading. 

m.           m. 

mm. 

m. 

mm. 

o  to    27 

0.0 

1  60 

-1.8 

28  to    47 

—  O.I 

170     • 

—  2  .1 

48  to    60 

—  0.2 

180 

—  2-3 

61  to    72 

-0-3 

190 

-2.6 

73  to    8  1 

-0.4 

200 

-2.8 

82  to    90 

-0-5 

2IO 

-3-0 

91  to    98 

-0.6 

2  2O 

-3-3 

99  to  105 

-0.7 

230 

-3-7 

106  to  112 

-0.8 

240 

—4.0 

113  to  118 

-0.9 

250 

-4-3 

119  to  124 

—    .0 

260 

-4-7 

125  to  130 

—    .1 

270 

-5-o 

131  to  136 

—      .2 

280 

-5-4 

137  to  141 

-    -3 

290 

-5-8 

142  to  146 

-    -4 

300 

-6.2 

147  to  150 

-    -5 

246        PRECISE  LEVELING  — TRIGONOMETRIC  LEVELING 


CORRECTION 


247 


TABLE  I.  —  DIFFERENTIAL  CORRECTION  FOR  CURVA- 
TURE AND  REFRACTION 


Mean 
length 
of  sight 
in  rod 
interval 
in  milli- 
meters. 

Difference  of  sights  in  rod  interval  in  millimeters. 

2 

4 

6 

s 

1C 

12 

14 

(6 

IS 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

4 

44 

46 

48 

50 

5~ 

54 

5658 

10 

20 
30 

40 
50 

60 
TO 
80 
90 

100 

no 

120 
130 
140 

ISO 

160 
170 
180 
190 
200 

210 
22O 
230 
240 
250 

260 
270 
280 
290 

300 

3io 
320 
330 
340 
350 

360 
400 
440 
480 
520 

.0 

-0 
.0 

c 

.0 

c 
c 
.c 
.0 

.0 

c 
.c 

.0 
.0 

.0 

.0 
.0 
.0 

.0 
.0 

.0 

0 

.0 
.0 
.0 
.0 

.0 
.0 
.0 
.0 

0 
.0 
.0 
.0 

.c 
c 
Q 

c 

~ 
,0 
.0 

c 

c 
.0 

c 
c 

Q 
0 
.0 
0 

0 
0 

0 
0 

c 

0 

0 
0 

0 
0 

0 

.0 

.0 

,c 

.0 

.0 
.0 

.0 
.0 

.0 

.c 

.0 

c 

0 

c 

.X 

.  I 
.X 

.1 
.X 

.X 
.1 

.1 
.1 

.1 
.1 

I 
I 

.1  .1 
.1  .1 

— 

— 

—  ^ 

I 

.1 
.X 
.1 
.1 

.1 

.1 

.1 

—  . 

.2 
.2 

.2 

.2 

.1 

.1 

«- 
.1 

,1 

.1 
.1 
_^_ 
.2 
.2 

.2 
.2 

.1  .1 
.1  .1 
.1  .1 
.1  .1 

.1  _£ 
.1   .2 
.2  .2 
.2  .2 
.2  .2 

.2  .2 
.2  .2 

.0 
.0 
.0 

.c 
.c 
.c 

.0 
.0 

.0 

.0 
.0 
0 

Q 

0 
.0 

.0 
.0 

z 
0 
.0 

-0 

.0 

0 

.0 
.0 

.0 

^0 

.0 

.0 

c 

.0 

.0 

— 

.1 
.1 
.1 

.1 

.1 

.1 
.1 

.1 

.X 

.1 
.1 

.1 

.1 

.1 
.X 

.1 
.1 

.1 
.1 
.X 
.X 
.X 

.1 

.X 

.c 
.c 

c 
I 

I 

.0 
I 

.1 
I 
.1 

.  I 
.1 

.1 
.1 
.1 

.1 

.1 

I 
.1 
.1 
.1 

I 

.1 

.1 
.1 

.1 

.1 
.X 

.1 
.1 
.1 

.1 
.1 

.1 

.1 

.1 

.1 

.1 
.1 
.1 

.1 
.1 

.1 
I 

I 
.1 

.1 

.1 
.1 
.1 
.1 

I 

.1 
.1 

.1 
.1 
.1 

.1 
.1 

2 
.2 

.1 

.1 

.1 
.1 

.1 

.1 

.2 

.2 
.2 

.1 
.1 

.1 
.1 

.1 

.c 
.c 
.c 
c 

.c 

.c 
,c 
.c 
.c 

.c 

.0 

c 

.0 

.0 

.0 

.0 

I 

I 

I 

.1 

.0 

0 

.0 

.c 
.0 
.c 
.0 

.0 

-C 

c 
.c 
.c 
.c 

.0 
.0 

.0 

.0 
.0 

.c 

.0 
.0 

1 

.c 

.0 

.c 
.0 
.0 
.0 

."6 

.0 

.0 

c 

I 
I 

I 

I 
I 

I 
I 

I 
I 

I 

I 

I 
I 
.1 

I 
I 

I 
I 
I 
I 

.  I 

I 
I 
I 

I 
I 
I 
I 

,  I 

I 
I 

I 
I 
I 
I 

.1 

.1 

I 
I 

I 

.1 

I 
I 
I 

.1 
.1 

I 
.1 

.1 

I 
.1 

;I 

.  I 

•' 

.1 

I 

.1 

.1 

" 

.2 

.2 

.2 

.2 

.2 

.2 

.2 

.2  .2 

.1 
.1 

.1 

.1 
.1 

.1 
.1 
.1 

.1 
.X 

I 

_I 
2 
2 
2 

2 

.1 

I 

-2 
2 

-2 
.2 

.2 
.2 

2 

3 
3 
3 
3 

3 

•3 

^ 

.2 
.2 

3 
.3 
•  3 
3 
.3 

•3 

_i 

-4 
4 
4 

-2  -3 
•3  -3 

3  -3 

3  -3 
3  -3 
3    3 
3  -3 

3,3 
.3J4 

4    4 
4  .4 
4  -4 

.1 

.2 
.2 
.2 

2 
2 

.2 

c 
.c 
.in 

.0 
.0 

.c 
z 
0 

.0 
.0 
.0 

.c 
0 

.c 

.c 
.c 

.c 
.c 

.c 

' 
z 
.0 

0 

— 

I 

.0 
.0 
.0 

.0 

c 

.1 

I 
I 

I 

2 
2 

2 
2 
2 

2 

— 

•3 

.2 
.2 

2 
.2 
2 

.2 
2 

2 

.2 

—  — 

•  3 
3 
3 

2 
.2 

.2 

-2 

3 
3 
3 

'2 
2 

2 

3 
3 
3 
3 

2 
2 

2 

— 

3 
3 
3 

3 

2 

2 

3 
3 
^ 

2 

3 

.3 

3 

} 

3 

3 
3 

.3 
•  3 

^ 

I 
I 
I 
.1 

I 

I 
_I 

2 

T 

2 
2 

2 

2 

2 

2 

.2 
.2 

2 
2 

2 

3 

3 
3 

_d 

4 

4 
4 

•  4 
•  4 

4 
4 

248        PRECISE  LEVELING  — TRIGONOMETRIC  LEVELING 


TABLE  J.  —  CORRECTION  FOR  TEMPERATURE   (IN 
MILLIMETERS) 


Temp.  C. 

Difference  of  elevation  in  meters. 

i 

2 

3 

4 

5 

6 

7 

8 

9 

IO 

ii 

12 

13 

14 

I 

O.O 

O.O 

O.O 

O.O 

O.O 

o.o 

O.O 

O.O 

O.O 

o.o 

o.o 

0.0 

0.0 

O.I 

2 

O.O 

o.o 

O.O 

o.o 

O.O 

o.o 

O.I 

O.I 

O.I 

O.I 

O.I 

O.I 

O.I 

O.I 

3 

o.o 

o.o 

O.O 

o.o 

O.I 

O.I 

O.I 

O.I 

O.'l 

O.I 

O.I 

O.I 

0.2 

0.2 

4 

o.o 

o.o 

O.O 

O.I 

O.I 

O.I 

O.I 

O.I 

O.I 

0.2 

O.2 

O.2 

O.2 

0.2 

5 

0.0 

o.o 

O.I 

O.I 

O.I 

O.I 

O.I 

O.2 

O.2 

O.2 

O.2 

O.2 

o-3 

0-3 

6 

o.o 

o.o 

O.I 

O.I 

O.I 

O.I 

O.2 

O.2 

O.2 

O.2 

0-3 

0-3 

o-3 

0-3 

7 

o.o 

0. 

O.I 

O.I 

O.I 

O.2 

O.2 

O.2 

O.2 

0-3 

0-3 

0-3 

0.4 

0.4 

8l 

0.0 

0. 

0. 

O.I 

O.2 

O.2 

O.2 

0-3 

0-3 

o-3 

0.4 

0-4 

0.4 

0.4 

9 

0.0 

O. 

0. 

O.I 

0.2 

O.2 

O.2 

0-3 

0-3 

0.4 

0.4 

0.4 

o-5 

0-5 

10 

o.o 

O. 

0. 

0.2 

0.2 

O.2 

0-3 

0-3 

0.4 

0.4 

0-4 

0-5 

0.6 

ii 

0.0 

0. 

0. 

O.2 

O.2 

0-3 

0.3 

0.4 

0.4 

0.4 

o-5 

o-5 

0.6 

0.6 

12 

0.0 

0. 

0. 

0.2 

0.2 

0.3 

o-3 

0.4 

0.4 

0-5 

o-5 

0.6 

0.6 

o-7 

13 

o.o 

0. 

0.2 

0.2 

0-3 

0.3 

0.4 

0.4 

o-5 

0-5 

0.6 

0.6 

0.7 

0.7 

14 

O.I 

0. 

0.2 

0.2 

0-3 

0.3 

0.4 

0-4 

0.6 

0.6 

o-7 

0.7 

0.8 

IS 

O.I 

O. 

0.2 

0.2 

o-3 

0.4 

0.4 

0-5 

0-5 

0.6 

0.7 

0.7 

0.8 

0.8 

16 

O.I 

0. 

0.2 

0-3 

o-3 

0.4 

0.4 

0-5 

0.6 

0.6 

0.7 

0.8 

0.8 

o-9 

17 

O.I 

O. 

0.2 

0-3 

0.3 

0.4 

0-5 

o-5 

0.6 

0.7 

0.8 

0.8 

0-9 

o-9 

18 

O.I 

O. 

O.2 

0-3 

0.4 

0.4 

0.6 

0.6 

o-7 

0.8 

o  -9 

°-9 

i  .0 

19 

O.I 

0.2 

O.2 

0-3 

0.4 

0.5 

°  5 

0.6 

0.7 

0.8 

0.8 

o-9 

.0 

i  .1 

20 

O.I 

O.2 

O.2 

0-3 

0.4 

o-5 

0.6 

0.6 

0.7 

0.8 

o-9 

I  .0 

.0 

i  .1 

21 

O.I 

O.2 

O.2 

o-3 

0.4 

o-5 

0.6 

0.7 

0.8 

0.8 

0-9 

i  .0 

.1 

I  .2 

22 

O.I 

O.2 

0-3 

0.4 

0.4 

o-5 

0.6 

0.7 

0.8 

0.9 

i  .0 

i  .1 

.1 

I  .2 

23 

O.I 

O.2 

o-3 

0.4 

0-5 

0.6 

0.6 

0.7 

0.8 

o-9 

I  .0 

i  .1 

.2 

1-3. 

24 

O.I 

0.2 

0-3 

0.4 

0-5 

0.6 

0.7 

0.8 

0.9 

i  .0 

i.i 

1.2 

.2 

25 

O.I 

O.2 

o-3 

0.4 

0-5 

0.6 

0.7 

0.8 

0-9 

I  .0 

i  .1 

1.2 

•3 

i-4 

26 
27 

O.I 
O    I 

O.2 
O    2 

o-3 

O     7 

0.4 

O   4 

o-5 

r\     f 

0.6 
0.6 

0.7 
0.8 

0.8 

o-9 

I    O 

I  .0 

i  .1 

I  .2 

•3 

I£ 

•  / 
28 
2Q 

O.I 
O  .  I 

O.2 
O  .  2 

w  -6 
O   A 

0.4 

Io   s 

^  •  3 

0.6 
o  6 

0.7 

0*7 

0.8 
0.8 

0.9 
/-i  c\ 

1  .0 

1  .1 

I  .2 

1-3 

•  4 

-  5 
1.6 

*y 

30 

O.I 

O.2 

w  .q. 

0.4 

u  •  o 

0.5 

0^6 

-  / 

0.7 

0.8 

u.y 
I  .O 

1  .1 

I  .2 

i-3 

i-4 

1.6 

31 

O.I 

0.2 

0.4 

0-5 

0.6 

0.7 

0-9 

I  .O 

1  .1 

I  .2 

i  -4 

r-5 

1.6 

i  -7 

32 

O.I 

0-3 

0.4 

o-5 

0.6 

0.8 

o-9 

I  .O 

I  .2 

1-3 

i  .4 

1  .5 

i  .7 

1.8 

33 

O.I 

o-3 

0.4 

o-5 

0.7 

0.8 

0.9 

I  .1 

I  .2 

i  .4 

1.6 

i  .7 

1.8 

34 

i  < 

O.I 
O    I 

0.3 

Ct     1 

0.4 

OA 

0-5 
0.6 

o-7 
f\  i-j 

0.8 
o  8 

i  .0 

I  .1 

I  .2 

i-4 

i-5 

1.6 

1.8 

i-9 

6  j 
36 

O.I 

u  -6 

o-3 

•4 

0.4 

0.6 

u-  / 
0.7 

\J  .  O 

0.9 

i  .0 

I  .2 

I  .A 

1.6 

i-7 

2  .  0 
2  .0 

37 

O.I 

0.3 

0.4 

0.6 

o-7 

o-9 

I.O 

1.2 

1-3 

I  .5 

1.6 

1.8 

1  -9 

2  .1 

38 

O.I 

0.3 

0.5 

0.6 

0.8 

0.9 

i  .1 

I  .2 

i-4 

I  .5 

i  .7 

1.8 

2  .0 

2.1 

39 

O.2 

0.5 

0.6 

0.8 

0.9 

i  .1 

I  .2 

i-4 

1.6 

1-7 

i  .9 

2  .O 

2  .2 

40 

O.2 

o-3 

°-5 

0.6 

0.8 

i  .0 

i  .1 

1-3 

i-4 

1.6 

1.8 

1.9 

2  .1 

2  .2 

41 

O.2 

0.3 

0.5 

0.7 

0.8 

I  .0 

i  .1 

1-3 

i  .5 

1.6 

1.8 

2.0 

2  .1 

2-3 

42 

0.2 

0.3 

0.5 

0.7 

0.8 

i  .0 

I  .2 

1.5 

1.7 

1.8 

2.0 

2.2 

2-3 

43 

0.2 

0.3 

0.5 

0.7 

0-9 

i  .0 

I  .2 

I  .4 

i  .5 

i  .7 

i-9 

2  .1 

2  .2 

2.4 

44 

O.  2 

0.3 

0.5 

0.7 

o-9 

i  .1 

1.2 

i-4 

1.6 

1.8 

2  .1 

2-3 

2-5 

45 

O.2 

0.3 

0.5 

•0.7 

0-9 

i.i 

1-3 

i-4 

1.6 

1.8 

2  .O 

2  .2 

2-3 

2-5 

DATUM  249 

164.  Datum. 

The  datum  for  precise  levels  is  mean  sea-level,  or  the  surface 
of  the  geoid,  as  found  from  tidal  observations.  This  is  assumed 
to  be  correctly  given  by  the  mean  of  the  several  "  annual  means  " 
as  derived  from  tidal  observations  for  sea-level.  The  heights  of 
the  tide  are  recorded  automatically  on  a  self-registering  gauge. 
(See  Cut.)  The  vertical  motion  of  the  float  is  reduced  (the  ratio 


FIG.  Qia.     Self-Registering  Tide  Gauge. 
(Coast  and  Geodetic  Survey.) 

depending  upon  the  range  of  tide)  by  passing  the  connecting 
wire  and  cord  over  a  series  of  pulleys,  and  is  communicated  to  a 
recording  pencil  which  marks  on  a  sheet  of  paper  passing  over  a 
revolving  drum.  The  drum  is  revolved  at  a  uniform  rate  by 
clock  mechanism.  The  height  of  the  water  is  referred  to  a  bench 
mark  in  the  vicinity.  Observations  of  the  tide  should  be  ex- 


250       PRECISE  LEVELING— TRIGONOMETRIC  LEVELING 

tended  over  a  period  of  at  least  one  year  in  order  to  determine 
sea-level  with  sufficient  precision  for  this  class  of  leveling.  In  the 
tidal  records  at  some  stations  there  appear  to  be  small  systematic 
variations  in  the  annual  means  extending  over  periods  of  several 
years;  but,  taking  the  records  as  a  whole,  the  variations  do  not 
seem  to  follow  any  particular  law,  and  they  are  treated  as  acci- 
dental. (See  Coast  Survey  Special  Publication  No.  26.) 

165.  Potential. 

In  order  to  investigate  the  nature  of  the  orthometric  correction, 
due  to  the  convergence  of  level  surfaces,  it  will  be  necessary  to 
consider  first  some  of  the  elementary  mechanical  principles  of  the 
earth's  gravitation  and  rotation. 

Whenever  two  attracting  bodies  are  separated,  work  is  done 
upon  them  and  energy  is  stored  up;  that  is,  the  potential  energy 
of  the  system  is  increased.  The  change  in  potential  energy  is 
measured  by  the  amount  of  work  done.  When  the  bodies  are 
an  infinite  distance  apart,  the  potential  energy  is  a  maximum ; 
when  the  bodies  are  in  contact,  the  potential  energy  of  the  system 
is  zero.  If  the  masses  are  free  to  move,  they  will  always  move 
in  such  a  direction  as  to  diminish  the  potential  energy  of  the 
system. 

If  we  imagine  a  unit  mass  placed  at  any  point  P  in  space  and 
attracted  by  a  mass  M,  and  if  the  potential  energy  of  the  unit 
mass  be  measured  by  the  work  done  upon  it  to  move  it  from  P  to 
infinity,  this  quantity  of  potential  energy  is  a  property  of  the 
given  point  P;  in  other  words,  it  is  a  function  of  the  coordinates 
of  P.  It  is  called  the  potential  at  that  point.  It  is  not  necessary 
that  there  should  actually  be  a  unit  mass  at  the  point,  but  the 
conditions  are  such  that  if  a  unit  mass  were  placed  at  P,  it  would 
have  this  amount  of  potential  energy.  It  should  be  observed 
that  the  increase  of  potential  energy  is  measured  by  the  fall  in 
potential. 

1 66.  The  Potential  Function. 

If  an  attracting  body  M  be  divided  into  small  elements,  and 
the  mass  Aw  of  each  element  be  divided  by  its  distance  from  a 


THE  POTENTIAL   FUNCTION  251 

point  P,  the  limit  of  the  sum  of  all  these  fractions,  as  the  elements 
are  made  smaller,  is  called  the  value  at  P  of  the  potential  function 
due  to  M,  or  simply  the  potential  of  P.  Calling  this  function  V, 
then 


of,  if  Aw  is  of  density  6  and  has  the  coordinates  x',  /,  z',  and  P 
has  the  coordinates  x,  y,  z,  then 


x'  -  *Y  +  (y'-  y)2  +  («'  -  z)2]* 

The  integration  over  the  entire  mass  gives  the  value  of  the  po- 
tential function  at  P.* 

167.  The  Potential  Function  as  a  Measure  of  Work  Done. 

The  amount  of  work  required  to  move  a  unit  mass  (concen- 
trated at  a  point)  from  a  point  PI  to  another  point  P2,  by  any 
path  (Fig.  92),  against  the  attraction  of  a  mass  M,  is  equal  to  the 
fall  in  potential  V\  —  F2,  where  Vi  and  F2  are  the  values  of  the 
potential  function  at  the  points  PI  and  P2.  To  show  this,  let  r\ 
and  r2  be  the  distances  from  the  center  of  M  to  the  points  PI  and 
P2. 


FIG.  92. 

The  work  done  in  moving  a  unit  mass  through  a  small  space  dr 
equals  the  force  (—  J  times  the  space  dr.     But  the  force  (^~\  at 

any  point  A  equals  —  —  at  that  point,  since  V  (for  a  unit  mass) 

=  - .    Hence  the  work 

r 

rdV     j        T/        T/  r       i 

=  -  Jri  *r •  Jr  =  Fl  -  F2;  [II9] 

that  is,  the  work  done  equals  the  fall  in  potential. 

*  See  Peirce,  Theory  of  the  Newtonian  Potential  Function. 


252        PRECISE  LEVELING— TRIGONOMETRIC  LEVELING 

If  the  point  P2  is  moved  to  an  infinite  distance,  F2  become  zero, 
and  the  potential  at  P\  then  equals  the  work  done  in  moving  the 
unit  mass  from  PI  to  infinity;  or  it  is  the  work  done  by  it  in 
moving  from  infinity  to  the  point  P\. 

1 68.   Equipotential  Surfaces. 

A  level  surface,  or  an  equipotential  surface,  is  one  having  at 
every  point  the  same  gravity  potential.  It  is  everywhere  per- 
pendicular to  the  direction  of  gravity.*  The  mean  surface  of 
the  ocean  is  such  a  surface.  The  surface  of  any  lake  is  also  an 
equipotential  surface.  From  the  proof  given  in  the  preceding 
article  it  is  evident  that  if  there  are  two  such  equipotential  sur- 
faces, the  difference  in  potential  is  the  work  done  upon  a  unit 
mass  in  moving  it  from  one  surface  to  the  other.  This  difference 
in  potential  is  independent  of  any  particular  points  on  the  sur- 
faces and  of  the  path  followed  in  passing  from  one  to  the  other; 
for  example,  the  work  done  in  raising  a  unit  mass  from  sea-level 
to  the  south  end  of  a  lake  is  the  same  as  the  work  done  in  raising 
a  unit  mass  from  sea-level  to  the  north  end  of  the  lake.  Since 
the  work  done  is  the  force  (w)  times  the  distance  (dti)  through 
which  it  acts,  it  is  evident  that  w  X  dh  is  a  constant  between  two 
level  surfaces.  Also,  since  g  varies  as  the  weight  (force),  g  X  dh 
is  a  constant  between  these  two  surfaces. 

The  force  of  gravity  is  less  at  the  equator  (Art.  144)  than  at  the 
poles,  on  account  of  the  action  of  the  centrifugal  force.  Hence 
we  should  expect  to  find  that  a  given  level  surface  is  farther  from 
sea-level  at  the  equator  than  it  is  at  a  point  nearer  the  pole.  If 
several  such  surfaces  be  drawn  (Fig.  93),  they  will  be  seen  to 
converge  toward  the  pole.  They  are  all  parallel  to  each  other 
at  the  equator  and  at  the  poles,  and  have  their  greatest  difference 
in  direction  at  <£  =  45°. 

Since  g  is  about  one-half  of  one  per  cent  less  at  the  equator  than 

*  It  may  be  proved  that  if  there  is  a  resultant  force  at  a  point  in  space  due  to 
attracting  masses,  this  force  acts  in  the  direction  of  the  normal  to  the  equipotential 
surface  through  the  point  (see  Peirce,  Theory  of  the  Newtonian  Potential  Function, 
p.  38).  It  should  be  kept  in  mind  that  the  "  force  of  gravity"  is  the  resultant  of 
the  force  of  attraction  and  the  centrifugal  force. 


EQUIPOTENTIAL  SURFACES 


253 


at  the  pole,  the  height  h  between  surfaces  is  about  one-half  of  one 
per  cent  greater  at  the  equator.  Hence,  if  a  level  surface  were 
icoo  meters  above  the  sea-surface  at  the  equator,  it  would  be 
only  995  meters  above  sea-level  at  the  pole.  A  surface  at  half 
the  elevation  would  converge  (very  nearly)  half  as  much.  In  the 
line  of  levels  run  from  San  Diego  to  Seattle  the  convergence  was 
found  to  be  about  i \  meters,  showing  that  at  high  elevations  this 
error  is  by  no  means  a  negligible  one  in  precise  leveling. 


E 


O 


FIG.  93. 


It  is  evident  that  if  a  series  of  bench  marks  is  established  along 
a  meridian  (in  the  northern  hemisphere),  and  all  are  placed  at 
the  same  elevation,  using  the  ordinary  methods,  those  at  the 
northern  end  of  the  line  lie  nearer  to  sea-level  than  those  at  the 
southern  end  of  the  line.  It  becomes  necessary,  then,  to  revise 
the  definition  of  elevation. 

If  the  ordinary  definition  of  elevation  is  retained,  and  no  allow- 
ance made  for  convergence  of  level  surfaces,  then  different  results 
for  the  elevation  of  a  point  will  be  obtained,  according  to  which 
path  is  followed.  If  we  measure  vertically  upward  from  A  to  B 
(Fig.  94),  and  then  level  by  means  of  the  water  surface  BC,  we 
obtain  a  greater  height  for  point  C  than  we  should  if  we  leveled 
by  water  from  A  to  D  and  then  measured  vertically  upward  from 
D  to  C.  If  a  correction  is  applied,  however,  to  allow  for  the  con- 
vergences of  these  surfaces,  the  result  is  that  different  portions 


254        PRECISE  LEVELING— TRIGONOMETRIC  LEVELING 

of  the  lake  surface  have  different  elevations,  which  is  apparently 
absurd  if  the  true  nature  of  the  level  surface  is  not  understood. 
In  order  to  avoid  this  apparent  difficulty  another  method  some- 
times employed  is  to  number  all  the  surfaces  with  a  serial  number 
(called  the  Dynamic  Number),  so  that  all  points  on  the  same 
surface  will  have  their  elevation  expressed  by  the  same  number. 
This  number  is  defined  as  the  work  required  to  raise  one  kilogram 
from  sea-level  to  the  given  surface,  the  unit  being  the  kilogram- 


FIG.  94. 

meter  at  sea-level  in  latitude  45°.  The  United  States  Coast 
Survey  has  adopted  the  method  of  applying  to  ordinary  elevations 
the  correction  for  convergence,  called  the  Orthometric  Correction. 
The  Standard  Elevations  of  the  Coast  Survey  in  Special  Pub- 
lication No.  1 8  are  given  by  the  Orthometric  Elevation. 

169.  The  Orthometric  Correction. 

Let  W  be  the  work  (in  absolute  units)  required  to  raise  a  unit 
mass  from  sea-level  to  a  point  at  elevation  h,  and  let  H  be  the 
dynamic  number  of  the  surface  through  the  point,  defined  by  the 
quotient  W  -f-  #45,  where  #45  is  the  value  of  g  at  sea-level  in 
latitude  45°  (Equa.  [p6a],  p.  210).  Then,  since  g  X  dh  is  constant 
for  two  level  surfaces  separated  by  height  dh, 


W 


—  I    gdh  =  £45  /    (i  —  0.002644  cos  2  <f>  .  .  .  ) 

«/o  t/o 


dh 


in  which  the  integration  takes  place  along  the  curved  vertical. 


THE  ORTHOMETRIC  CORRECTION 


255 


Integrating,   W  =  g4s  |(i  -  0.002644  cos  2  4>)  h  .  .  .  1  .    [120] 


The  dynamic  number 

W 

=  H  =  —  =  h  (i  —  0.002644  cos  2  0  .  .  .  ).          [121] 

To  find  the  correction  to  the  elevation  due  to  a  change  in  the 
latitude,  differentiate  the  last  equation  with  respect  to  </>  as  the 
independent  variable,  and  we  obtain 

o  =  dh  —  0.002644  (—  2  h  sin  2  <f>  d<f>  +  cos  2  <£  dh  .  .  .  ) 
=  dh(i  —  0.002644  cos  2  <£)  +  0.005288  /r  sin  2  0  J<£, 

0.005288 /^  sin  2  0  d<f>  T      -, 

and  dh  =  — —  [.122] 

i  —  0.002644  cos  2  0 

=  —(0.005288  h  sin  20)  (1+0.002644  cos  2<£  .  . .  )^arci',*  [123] 
the  factor  arc  i'  being  introduced  to  reduce  d<j>  to  minutes  of  arc. 

A  more  definite  idea  of  the  magnitude  of  this  correction  may 
be  gained  from  the  following  example.  Assuming  that  the  ele- 
vation of  Lake  Michigan  is  177  meters  at  Chicago,  latitude 
41°  53',  what  is  the  elevation  of  the  lake  at  Milwaukee,  in  latitude 
43°  °3'?  In  tne  formula,  h  =  177™,  d<i>  =  70',  and  <j>  =  42°  28'; 
the  computed  values  of  dh  is  —0.0190™,  and  the  lake  level  at  Mil- 
waukee is  therefore  176.9810  meters.  Tables  for  computing  the 
orthometric  correction  will  be  found  in  Coast  Survey  Special 
Publication  No.  18,  pp.  54-56. 

The  relation  between  the  dynamic  numbers  and  the  ortho- 
metric  elevations  is  illustrated  in  the  following  table,  which  is  an 
extract  from  the  special  publication  just  mentioned. 


Station. 

Latitude. 

Orth.  elev  meters. 

Dyn.  number. 

Smithland,  La  

0             / 

10   ">"> 

14.7729 

14.7545 

Meridian  Miss 

•72    22 

IO4.   Q4Q4 

104.8292 

Amblersburg,  W.  Va  
Summit,  Cal. 

39  23 
24.  20 

494.9221 
Il6^  434^ 

494.6287 
1164.1008 

Riordan,  Ariz.                   ... 

3<j  13     * 

22l6.  l>4'>2 

2213  .8lI2 

*  For  additional  terms,  neglected  in  the  above  formula,  see  Coast  and  Geodetic 
Survey  Special  Publication  No.  18,  p.  49.  See  also  Ch.  Lallemand,  Nivellement 
ie  Haute  Precision,  Encyclopedic  des  Travaux  Publics,  Paris,  1912. 


256        PRECISE  LEVELING— TRIGONOMETRIC  LEVELING 


170.  The  Curved  Vertical. 

In  view  of  what  has  been  said  regarding  the  change  in  the 
direction  of  level  surfaces  with  an  increase  in  elevation,  it  is  clear 
that  the  vertical  line  is  curved,  being  concave  toward  the  pole, 
and  therefore  that  any  observation  for  latitude  made  at  a  point 
above  sea-level  is  referred,  not  to  the  true  normal  to  the  surface 
at  sea-level,  but  to  the  direction  of  that  portion  of  the  vertical 
which  is  at  the  elevation  (h)  of  the  station.  In  order  to  deter- 
mine the  amount  of  the  correction  to  reduce  the  observed  latitude 


FIG.  95. 

to  its  value  at  sea-level,  refer  again  to  Equa.  [122],  p.  255.  An 
inspection  will  show  that  the  denominator  of  this  fraction  is 
usually  not  far  from  unity;  and  since  the  correction  desired  is 
itself  quite  small,  we  may  assume 

dh  =  —  0.005288  h  sin  2  <£  d<f>.  [124] 

The  correction  to  the  observed  latitude  is  the  difference  in  the 
slope  of  the  two  surfaces  (sea-level  and  the  level  of  station) 
measured  in  the  plane  of  the  meridian.  From  Fig.  95  it  is  seen 
that  the  angle  between  the  level  surface  through  S  and  a  surface 
parallel  to  sea-level  drawn  through  S  is  dh  -f-  Rd  <£.  But,  by 
Equa.  [124], 

dh  0.005288  h  sin  2  <f> 

Rd<f>~  R 

Reducing  this  to  seconds  of  arc, 

dh         _  0.005  2 88 /?  sin  2  <fr 

R  d<t>~  R&rci" 


REDUCTION  TO  STATION  MARK  257- 

Since  R  arc  i"  =  101.3  feet  (very  nearly),  the  correction  to  the 
latitude  may  be  written 

—  0^.0522  h  sin  2  <f>t  [125] 
where  h  is  in  thousands  of  feet;  or,  if  h  is  in  meters,  the  correction  is 

—  0.000171  h  sin  2  <£.  [126] 

171.  Trigonometric  Leveling. 

The  method  of  measuring  the  vertical  angles  between  triangu- 
lation  stations  has  already  been  described  in  the  chapter  on  field- 
work.  From  the  field  note-book  we  have  the  several  measures 
of  the  angles,  the  height  of  the  instrument,  and  also  of  the  point 
sighted  in  each  case  above  the  station  marks.  The  elevation  of 
one  station  above  sea-level  is  assumed  to  be  known,  and  that  of 
the  other  is  to  be  computed.  Before  this  can  be  done,  the  angle 
must  be  reduced  to  the  value  it  would  have  if  the  instrument  and 
the  point  sighted  were  coincident  with  the  station  marks. 

172.  Reduction  to  Station  Mark. 

From  the  diagram  (Fig.  96)  it  is  evident  that  if  i  is  the  height 
of  the  instrument  at  A ,  and  o  that  of  the  object  sighted  at  B,  and 


FIG.  96. 


5  the  distance  between  stations,  obtained  from  the  triangulation, 
then  the  correction  to  the  vertical  angle  at  A  is 


Four  places  in  the  logarithms  are  sufficient  in  computing  this 
correction. 


258        PRECISE  LEVELING— TRIGONOMETRIC  LEVELING 

This  reduction  need  be  made  only  in  case  of  reciprocal  obser- 
vations, that  is,  observations  of  the  vertical  angle  from  both  ends 
of  the  line.  In  case  of  observations  from  one  station  only,  the 
quantity  i  —  o,  in  meters,  can  be  applied  directly  to  the  com- 
puted difference  in  elevation. 

When  a  sight  is  taken  from  one  station  PI  to  another  station 
P2,  the  verticals  of  the  two  stations  do  not  (in  general)  intersect, 
because  they  lie  in  different  planes.  If  we  imagine  a  plane  which 
is  parallel  to  both  verticals,  and  then  project  both  verticals  onto 
this  plane,  we  obtain  the  result  shown  in  Fig.  97. 


FIG.  97. 

173.  Reciprocal  Observations  of  Zenith  Distances. 

In  Fig.  97,  PI  and  P2  represent  the  two  instrument  stations; 
their  elevations  above  sea-level  are  PiSi  =  hi  and  P252  =  fe. 
The  ray  of  light  is  assumed  to  take  the  form  of  a  circular  curve, 


RECIPROCAL  OBSERVATIONS  OF  ZENITH  DISTANCES      259 

whose  radius  is  determined  by  the  coefficient  used  in  the  calcu- 
lation. The  two  measured  zenith  distances  are  ft  and  fo. 

The  angle  of  refraction  is  Af  =  rPiP2  =  TP2Pi  =  m  9, 
where  m  is  the  coefficient  of  refraction,  and  0  the  central  angle 
PiOP2.  The  radius  of  curvature  of  the  section  SiS2  is  Ra}  ap- 
proximately equal  to  OSi,  or  to  062  . 

The  quantity  to  be  computed  is  the  difference  in  elevation 
fa  —  hi,  which  may  be  found  by  solving  the  triangle  PiP^.* 

In  the  triangle  PiP^,  P*L*  =  fa  —  hi,  the  desired  difference 
in  elevation;  P\L<i  is  the  chord  joining  the  two  verticals  at  the 
level  surface  through  PI.  Observing  that  P\M  =  (Ra  +  hi) 

n  n 

sin  -  ,  and  PiLz  =  2  (Ra  +  hi)  sin  -  ,  we  have,  by  applying  the 
law  of  sines, 

fc^;...A+w*jx*^g.         w 

But  in  the  triangle 


=  (90°  -  *)  -  (180°  -  ft  -  Af) 


=  -90°- 


Also,  P2P,L2  =  180°  -    ri  +  Af  +  90°  - 

=  90°_fl_Af  +  l 
Adding  (f)  and  (g)  and  dividing  by  2, 


(h) 


2 

In  the  triangle  PiP2P2 

PiP^  =  180°  -  [9  +  (180°  -  fi  - 

=  -  0  +  fi  +  Af.  (i) 

*  The  following  formulae  are  those  adopted  by  the  Coast  and  Geodetic  Survey 
in  1915  (see  Special  Publications  Nos.  26  and  28). 


260        PRECISE  LEVELING  —  TRIGONOMETRIC  LEVELING 

Also,  PxPzLz  =  180°  -ft-  Ar.  (/') 

Adding  (i)  and  (j)  and  dividing  by  2, 

P.PA-Qo'-g  +  kf*)-  (*) 

Substituting  (A)  and  (&)  in  (e), 


(Q 


2  2 

Expanding  the  denominator  and  dividing  the  numerator  and 


denominator  by  cos  (-  --  ^)cos-,  we  obtain 

\          2       /  2 


2  (tf  «  +  Ai)  tan  -  tan  f^1—  ^ 

T      _^     =  _  2  \        2 


i  -tan-  tan 


2  2 

13 

Expanding  tan  -  in  series  (see  p.  330),  retaining  but  two  terms  of 

2 

the  series,  and  putting  6  =  —  , 


I \  l\    -       I  _     T»    9  I  I    ~      l  -     7~>  L-^Oj 


.j5.C,  [129] 

in  which 

^  =  i+f, 

^a 

the  correction  for  elevation  of  the  station  of  known  elevation, 


WHEN  ONLY  ONE  ZENITH  DISTANCE  IS  OBSERVED       261 
the  correction  for  the  difference  in  elevation, 

and  C  =  i  H  --  :  —  , 

12  Ra2 

the  correction  for  distance. 

The  logarithms  of  A  ,  B,  and  C  are  given  in  Tables  K,  L,  and  M,  for 

the  arguments  hi,  log  5  tan  —      -  ,  and  log  5,  respectively. 

174.  When  only  one  Zenith  Distance  is  Observed. 
From  (g)  and  (ti)  we  have 

f 

The  refraction  angle  is  Af  =  m8,  where  m  is  the  coefficient,  to 
be  obtained  from  the  best  obtainable  values,  and  which  is  ap- 
proximately equal  to  0.071  ;  substituting  mB  in  the  above  equation 
we  have 


and      tan          -      =  tan  (90°  +  (0.5  -  «)  9"  arc  i"  -  f  ,) 


since  0"  = 


Putting  this  5  term  =  k,  we  have 

tan  p^1)  =  tan  [90°  +  k  -  ft],  (n) 

Substituting  in  [129]  from  («), 

h,  -  h  =  s  tan  [90°  +k-h]A.B-C,  [130] 

in  which  A,  B,  and  C  have  the  same  meaning  as  before,  except 
that  B  is  given  for  the  argument  log  [s  tan  (90°  +  k  —  ft)]. 

Example.  Zenith  Distance  of  Mt.  Blue  from  Farmington,  87°  of  i8".8;  dis- 
tance, 15,519  meters;  m  =  0.071;  instrument  2.20  meters  above  station  mark;  point 
sighted  4.40  meters  above  station  mark;  elevation  of  Farmington,  181.20  meters. 


262        PRECISE  LEVELING  — TRIGONOMETRIC  LEVELING 


m  0.071 
(0.5— w)  0.429 


log  Ra 
"  sini' 


log  =  9.6325 

log  ^  =  4-1909 

colog  Ra  sin  i"  =  8. 5092 

2.3326 

6.8052  K 

4.6856 
i . 4908 


90  oo  oo 

215".!     =         Q3'35".i 
f  =  87°  07'  18  .8 
+     2°56/i6//.3 


tan 
log  s 

A 
B 
C 


Red.  to  Sta. 
Diff.  Eleva. 


796.51  meters 

2.20          " 


794- 31 
Elev.  Farmington  181 .  20 

Elev.  Mt.  Blue      975.51 
TABLE  K* 


hi. 

Log  A, 
units  of 
fifth 
place  of 
decimals. 

ft* 

Log  A, 
units  of 
fifth 
place  of 
decimals. 

•      *L 

Log  A, 
units  of 
fifth 
place  of 
decimals. 

ftfr 

Log  A, 

units  of 
fifth 
place  of 
decimals. 

Meters. 

Meters. 

Meters. 

Meters. 

O 

IS4I 

3IS6 

4770 

O 

II 

22 

33 

73 

1688 

3303 

4917 

I 

12 

23 

34 

220 

1835 

3449 

5064 

2 

J3 

24 

35 

367 

1982 

3596 

5211 

3 

14 

25 

36 

SH 

2128 

3743 

5357 

4 

IS 

26 

37 

661 

2275 

3890 

5504 

5 

16 

27 

38 

807 

2422 

4036 

5651 

6 

17 

28 

39 

954 

2569 

4183 

5798 

7 

18 

29 

40 

IIOI 

2715 

4330 

5945 

8 

19 

30 

4i 

1248 

2862 

4477 

6091 

9 

20 

31 

1394 

3009 

4624 

10 

21 

32 

i54i 

3156 

4770 

*  In  these  tables  log  Ra  is  taken  as  6.80444, 
Spheroid  of  1866. 


the  mean  radius  in  latitude  40°  on 


WHEN  ONLY  ONE  ZENITH  DISTANCE  IS  OBSERVED      263 

Table  K  gives  the  values  of  log  A ,  the  correction  factor  for  the 
elevation  of  the  known  station,  by  showing  the  limiting  values 
of  the  elevation  fe,  between  which  log  A  may  be  taken  as  o,  i, 
2,  3,  etc.,  units  of  the  fifth  place  of  decimals.  Log  A  is  positive, 
except  in  the  very  rare  case  where  hi  corresponds  to  a  point 
below  mean  sea-level. 

TABLE  L 


Log  5  tan  i 

Log  5  tan  i 

Log  s  tan  J 

(T«  -  Ti)  or  log 

Log  B,  units 

(T,  -  TO  or  log 

Log  B  units 

(Ti  -  Ti)  or  log 

Log  B  units 

5  tan  (90°  +  * 

of  fifth  place 

s  tan  (90°  +  k 

of  fifth  place 

5  tan  (90°  +  * 

of  fifth  place 

-r  i)  •  (*  in 

of  decimals. 

-  f  i)  •  (*  in 

of  decimals. 

-  Ti)  '  (s  in 

of  decimals. 

meters.) 

meters.) 

meters.) 

o 

2.167 

3-397 

3.685 

i 

9 

17 

2.644 

3-445 

3-7II 

2 

10 

18 

2.866 

3-489 

3-735 

3 

II 

19 

3.011 

3.528 

3.758 

4 

12 

IO 

3.121 

3  .565 

3-779 

5 

13 

21 

3.208 

3-598 

3-8oo 

6 

14 

22 

3.281 

3-629 

3.820 

7 

IS 

23 

3-343 

3-658 

3-839 

8 

16 

24 

3-397 

3-685 

3-857 

Table  L  gives  the  values  of  log  B,  the  correction  factor  for 
approximate  difference  of  elevation  by  showing  the  limiting 
values  of  log  [s  tan  J  (ft  -  ft)]  or  log  rs  tan  (90°  +  k  -  ft)]  be- 
tween which  log  B  may  be  taken  as  o,  i,  2,  3,  etc.,  units  of  the 
fifth  place  of  dec'mals.  Log  B  has  the  same  sign  as  the  angle 
i  (ft  -  ft)  or  90°  +  k  -  ft;  for  example,  if  log  [s  tan  £  (ft  -ft)] 
lies  between  3.565  and  3.598  and  J  (ft  —  ft)  is  positive,  logB  = 
+0.00013,  but  if  \  (ft  —  ft)  is  negative  then  log  B  =  —0.00013, 
i.e.,  9.99987  —  10,  the  former  way  of  writing  being  usually  more 
convenient  in  practice. 


264        PRECISE  LEVELING  — TRIGONOMETRIC  LEVELING 

TABLE  M 


Logs  (s  in  meters). 

Log  C,  units  of  fifth 
place  of  decimals. 

Logs  (sin  meters). 

Log  C,  units  of  fifth 
place  of  decimals. 

o.ooo 

5-297 

0 

4 

4.875 

5-352 

I 

5 

5-"3 

5-395 

2 

6 

5.224 

5-432 

3 

7 

5-297 

5-463 

Table  M  gives  the  value  of  log  C,  the  correction  factor  for  dis- 
tance between  stations,  by  showing  the  limiting  values  of  log  5 
between  which  log  C  may  be  taken  as  c,  i,  2,  3,  etc.,  units  of  the 
fifth  place  of  decimals.  Log  C  is  always  positive. 

PROBLEMS 

Problem  i.  Calculate  the  orthometric  correction  for  a  line  extending  2°  north- 
ward from  a  point  in  latitude  45°  N  at  an  elevation  of  1000  meters. 

Problem  2.  Compute  the  correction  for  reducing  to  sea-revel  a  latitude  observed 
at  an  elevation  of  one  mile  in  latitude  45°  N. 

Problem  3.  Vertical  angle  from  S  to  B,  +2°  24'  58".Q4.  Vertical  angle  from 
B  to  S,  —2°  35'  34". 20.  Elevation  of  S  =  108.87  meters;  distance,  23,931.6 
meters;  log  R0,  6.8052.  Compute  the  elevation  of  B. 


CHAPTER  XI 
MAP  PROJECTIONS 

175.  Map  Projections. 

Whenever  we  attempt  to  represent  a  spherical  or  a  spheroidal 
surface  on  a  plane  some  distortion  necessarily  results,  no  matter 
how  small  may  be  the  area  in  question.  The  problem  to  be 
solved  in  constructing  topographic  or  hydrographic  maps  is  to 
find  a  method  which  will  minimize  this  distortion  under  the 
existing  conditions.  The  number  of  projections  which  have 
been  devised  is  very  great;  for  the  description  and  the  mathe- 
matical discussion  of  the  properties  of  these  projections  the 
reader  is  referred  to  such  works  as  Thomas  Craig's  Treatise  on 
Projections,  United  States  Coast  and  Geodetic  Survey,  1882;  The 
Coast  and  Geodetic  Survey  Report,  1880;  C..L.  H.  Max  Jurisch, 
Map  Projections,  Cape  Town,  1890;  G.  James  Morrison,  Maps, 
Their  Uses  and  Construction,  London,  1902;  and  A.  R.  Hinks, 
Map  Projections,  Cambridge,  1912. . 

In  this  chapter  we  shall  consider  only  those  projections  which 
are  used  for  such  maps  and  charts  as  are  of  importance  in  geo- 
detic surveys  and  in  navigation. 

176.  Simple  Conic  Projection. 

In  this  projection  the  map  is  conceived  to  be  drawn  on  the 
surface  of  a  right  circular  cone  which  is  tangent  to  the  sphere 
or  the  spheroid  along  the  middle  parallel  of  latitude.  The  apex 
of  the  cone  lies  in  the  prolongation  of  the  axis  of  the  spheroid. 
From  Fig.  98  it  is  evident  that  the  distance  TA  from  the  apex 
to  the  parallel  through  A  is  equal  to  N  cot  0.  If  the  cone  is 
developed  on  a  plane  surface  we  shall  have  a  sector  whose 
center  is  T  and  whose  radius  is  N  cot  <f>.  (Fig.  99.)  All  other 
parallels  of  latitude  on  the  map  will  be  circles  drawn  about 

265 


266 


MAP  PROJECTIONS 


the  same  center  T,  and  all  meridians  will  be  represented  by 
straight  lines  passing  through  T.  The  spacing  between  the 
parallels  of  latitude  is  obtained  by  laying  off  distances  along 
the  central  meridian  which  are  proportional  to  the  distances 
between  the  same  parallels  on  the  spheroid.  The  position  of 
the  meridians  is  found  by  subdividing  the  middle  parallel  into 
spaces  which  are  proportional  to  the  lengths  of  the  arcs  of  the 


FIG.  98. 


same  parallel  on  the  spheroid.  Straight  lines  are  then  drawn 
from  the  center  T  through  these  points  of  sub-division.  Any 
meridian  or  any  parallel  may  be  assumed  for  the  central  meridian 
and  middle  parallel  of  the  map.  It  is  evident  from  the  above 
that  this  is  not  a  true  projection,  that  is,  the  points  are  not  those 
that  would  be  obtained  by  projecting  from  the  center  of  the 
sphere  onto  the  cone.  If  the  scale  of  the  map  is  such  that  the 
position  of  the  center  T  cannot  be  represented  on  the  paper, 
the  curves  may  be  laid  off  by  plotting  certain  points  by  means 
of  their  rectangular  coordinates  as  described  later  under  the 
poly  conic  projection. 

It  is  evident  that  the  meridians  and  parallels  of  a  conic 
projection  intersect  at  right  angles  in  all  parts  of  the  map,  as 
they  do  on  the  sphere.  The  scale  of  the  map  is  not  correct, 


BONNE'S  PROJECTION 


267 


however,  except  along  the  middle  parallel.  For  a  map  having 
a  great  extension  in  the  longitude  and  but  little  in  the  latitude^ 
the  conic  projection  is  fairly  accurate.  Fig.  100  shows  a  com- 
pleted conic  projection  covering  the  area  of  the  United  States. 


110°      105°      100°      95°       90°      85° 

FIG.  100.     Simple  Conic  Projection. 


80 ; 


75 ' 


177.  Bonne's  Projection. 

This  projection  is  a  modification  of  the  simple  conic  and  meets 
the  objection  that  the  scale  of  the  latter  becomes  inaccurate  as 
the  distance  from  the  middle  parallel  increases.  The  parallels 
of  latitude  are  concentric  circles  as  before,  but  each  parallel  is 
sub-divided  into  spaces  which  are  proportional  to  the  corre- 
sponding spaces  on  that  parallel  on  the  spheroid.  The  central 
meridian  and  all  parallels  are  therefore  correctly  sub-divided. 
The  meridians  are  obtained  by  joining  the  points  of  sub-division 
on  the  parallels.  The  meridians  in  this  projection  are  all 
curved,  except  the  central  one,  and  they  intersect  the  parallels 


268 


MAP   PROJECTIONS 


nearly,  but  not  quite,  at  right  angles  (Fig.  101).  The  distortion 
in  this  projection  is  very  small,  and  for  small  areas  it  is  practi- 
cally a  perfect  projection.  It  has  been  much  used  in  Europe. 


130°    125° 


115°   110°  105°   100°   95°    90°   85°    80°    75°    70°    66°    60° 


120° 


105°      100°      95°      90°      85° 

FIG.  101.     Bonne's  Projection. 


80° 


70° 


178.  The  Polyconic  Projection. 

The  idea  of  using  several  cones,  or  the  polyconic  projection, 
is  due  to  Mr.  F.  R.  Hassler,  the  first  superintendent  of  the  Coast 
Survey.  Each  parallel  of  latitude  shown  on  the  map  is  de- 
veloped on  a  cone  tangent  along  that  parallel.  The  radius 
(TA)  for  any  parallel  (latitude  0)  is  Ncot^r,  and  the  angle 
between  two  elements  of  the  cone  when  developed  is  approxi- 
mately 6  =  (d\)  sin  $,  as  will  be  evident  from  Fig.  102. 

In  constructing  the  map  the  degrees  of  latitude  are  laid  off 
along  the  central  meridian,  the  spacing  corresponding  to  the 
distances  on  the  spheroid.  The  points  where  the  meridians 
intersect  the  parallels  are  plotted  from  their  rectangular  co- 


THE   POLYCONIC  PROJECTION 


269 


ordinates,  the  coordinate  axes  being  in  each  case  the  central 
meridian  and  a  line  at  right  angles  to  it  drawn  through  the 
latitude  in  question.  The  coordinates  themselves  are  found 
as  follows:  In  Fig.  103,  let  A  be  the  intersection  of  some  meridian 
and  parallel  which  are  to  be  drawn  on  the  map.  Then  the 


FIG. 


FIG.  103. 


radius  TA  =  N  cot  <f>  may  be  computed  from  the  known  lati- 
tude of  A,  and  the  angle  6  may  be  computed  from  the  known 
difference  in  longitude  between  O  and  A  by  the  equation  6  = 
(d\)  sin  0.  Then  for  x  and  y  we  have 


and 


TA  sin  0  =  N  cot  <£  sin  (d\  sin 
y  =  TAversO  =  - — :vers0 


sin  & 


ztan- 
2 


=  x  tan  \  (d\  sin 


[132] 


Values  of  these  numbers  will  be  found  in  Tables  XVI  and  XVII. 


270 


MAP  PROJECTIONS 

°8  °8 


LAMBERT'S  PROJECTION  271 

It  is  evident  that  the  parallels  and  meridians  do  not  intersect 
at  right  angles  except  at  the  central  meridian.  The  meridian 
and  parallels  are  both  curved,  as  in  Bonne's  projection,  but 
since  the  lower  parallels  are  flatter  there  is  a  separation  of  the 
parallels  which  becomes  more  marked  toward  the  east  and 
west  margins  of  the  map.  For  this  reason  this  map  becomes 
less  and  less  accurate  as  the  longitude  is  extended.  In  mapping 
areas  which  extend  principally  north  and  south,  it  is  superior 
to  other  projections.  It  is  in  general  use  in  the  United  States 
for  Government  maps.  Fig.  104  shows  a  polyconic  projection 
covering  the  area  of  the  United  States. 

There  is  one  disadvantage  in  the  Polyconic  and  the  Bonne's 
projections,  namely,  that  if  two  maps  of  adjoining  areas  are 
to  be  placed  side  by  side  they  cannot  be  placed  exactly  in  con- 
tact because  the  limiting  (common)  meridian  curves  in  opposite 
directions  on  the  two  maps.  In  the  simple  conic  and  in  the 
Lambert  projection,  to  be  described  in  the  next  article,  the 
meridians  are  straight  and  this  difficulty  does  not  exist. 

179.  Lambert's  Projection. 

The  Lambert  projection  having  two  standard  parallels  was 
invented  about  the  middle  of  the  eighteenth  century,  but  has 
recently  been  brought  into  prominence  through  its  use  in  the 
French  battle  maps.  The  fundamental  notion  is  that  of  a  cone 
tangent  along  the  middle  parallel  of  the  map,  the  radius  of  this 
parallel  (on  the  map)  being  N  cot  <£,  and  the  angle  between 
the  central  meridian  and  any  other  meridian  being  (d\)  sin  0. 
This  would  give  a  map  in  which  one  parallel,  and  only  one,  is 
correctly  divided.  We  may,  however,  modify  the  projection 
so  as  to  have  two  standard  (correct)  parallels.  This  is  done  by 
reducing  the  scale  (multiplying  by  a  constant)  and  is  practi- 
cally equivalent  to  employing  a  cone  which  cuts  the  spheroid  in 
the  two  standard  parallels. 

The  other  parallels  are  so  spaced  that  the  scale  of  the  map  is 
the  same  for  all  azimuths  at  any  one  place,  that  is,  the  scale 
along  a  meridian  is  the  same  as  the  scale  in  an  east  and  west 


272  MAP   PROJECTIONS 

plane.  A  projection  having  this  property  is  said  to  be  "con- 
formal."  It  may  be  proved  that  this  condition  is  true  if  the 

spacing  between  parallels  is  0  H ,  where  /3  is  the  arc  of  the 

6  p<f 

meridian  between  parallels  on  the  original  tangent  cone  meas- 
ured from  the  parallel  of  contact,  and  p0  is  the  mean  radius  of 
curvature  of  the  spheroid  at  a  point  on  this  tangent  parallel. 
Since  the  projection  is  conformal,  all  lines  on  the  map  cut  each 
other  at  the  same  angles  as  do  the  corresponding  lines  on  the 
spheroid.  There  is  a  tendency,  therefore,  for  small  figures  to 
have  the  same  shape  on  the  map  that  they  have  on  the  earth's 
surface.  The  scale  of  this  map  is  correct  on  the  two  standard 
parallels.  Between  these  two  parallels  the  scale  is  a  little  too 
small  and  outside  these  parallels  the  scale  is  too  large.  The 
error  is  not  serious,  however,  if  the  standard  parallels  are  chosen, 
as  is  usual,  one  sixth  and  five  sixths  the  length  of  the  meridian 
arc  to  be  shown.  Fig.  105  shows  a  Lambert  projection. 


<>00_809  J70°  eO'SO'tp^gplO0  0°  10°  20°  30°    40° 


70°  60°  60°  40°  30°  20°  10°  0°  10°  20° 

FIG.  105.     Lambert  Projection. 

This  projection  may  be  extended  indefinitely  in  an  east  and 
west  direction  without  error.  The  error  becomes  greater  and 
greater  as  the  map  is  extended  to  the  north  and  south.  In 
this  respect  it  is  just  the  contrary  of  the  Polyconic  Projection. 


THE  GNOMONIC  PROJECTION 


273 


For  a  complete  description  of  this  projection,  together  with 
tables  for  projecting  maps,  see  United  States  Coast  Survey 
Special  Publications  47  and  52. 

1 80.  The  Gnomonic  Projection. 

In  the  gnomonic,  or  central,  projection  the  projecting  point 
is  at  the  center  of  the  sphere  and  the  plane  of  the  map  is  tangent 
to  the  sphere  at  some  selected  point.  Every  plane  through 
the  center  cuts  the  sphere  in  a  great  circle  and  cuts  the  map  in 
a  straight  line;  hence  every  great  circle  is  represented  by  a 
straight  line  and  every  straight  line  on  the  map  must  represent 
a  great  circle. 

Fig.  106  shows  the  Atlantic  Ocean  projected  on  a  plane 
tangent  at  <j>  =  30°  N  and  X  =  30°  W. 


70°         60°       60°      40°     30°     20°      10°       0°         10° 

FIG.  106.    Gnomonic  Projection  or  Great-circle  Chart. 

The  meridians  and  the  equator  are  of  course  represented  by 
straight  lines.  The  parallels  of  latitude  are  conic  sections,  in 
this  case  hyperbolas.  The  parallels  are  best  constructed  by 
employing  the  equations  of  the  curves  and  plotting  points  by 
means  of  coordinates. 


274  MAP  PROJECTIONS 

The  gnomonic  projection  is  used  almost  exclusively  for  deter- 
mining  the  positions  of  great  circles  for  the  purposes  of  naviga- 
tion. By  joining  any  two  places  by  a  straight  line  the  great- 
circle  (or  shortest)  track  is  at  once  shown.  The  latitudes  and 
longitudes  of  any  number  of  points  on  this  track  may  be  read 
off  the  chart  and,  if  desired,  may  be  transferred  to  any  other 
chart  and  the  curve  sketched  in.  The  point  where  the  great 
circle  approaches  most  nearly  to  the  pole  is  found  at  once  by 
drawing  from  the  pole  a  line  perpendicular  to  the  track.  The 
foot  of  this  perpendicular  is  the  vertex,  or  point  of  highest  latitude. 

181.  Cylindrical  Projection. 

If  a  cylinder  is  circumscribed  about  a  sphere  so  as  to  be  tan- 
gent along  the  equator,  and  if  points  be  projected  onto  the 
cylinder  by  straight  lines  from  the  center,  the  cylinder,  when 
developed  will  give  a  map  in  which  the  meridians  and  parallels 
are  all  straight  lines,  the  relative  distances  between  points  being 
approximately  correct  near  the  equator  but  distorted  in  high 
latitudes.  The  meridians  will  all  be  parallel  to  each  other.  The 
parallels  of  latitude  will  be  parallel  to  each  other  and  will  be 
spaced  wider  and  wider  apart  as  the  latitude  increases.  Evi- 
dently the  scale  of  the  map  is  different  for  different  latitudes. 
It  is  also  true  that  at  any  point  the  scale  along  a  meridian  is 
not  the  same  as  the  scale  along  a  parallel.  Such  a  projection 
is  of  no  practical  value,  but  it  aids  in  understanding  the  Mer- 
cator  chart  which  is  described  in  the  next  article. 

182.  Mercator's  Projection. 

A  modification  of  the  above  projection,  known  as  Mercator's, 
consists  in  so  spacing  the  parallels  of  latitude  that  the  relation 
between  increments  of  latitude  and  longitude  on  the  chart  is 
the  same  as  the  relation  between  increments  of  latitude  and  longi- 
tude at  the  corresponding  point  on  the  earth's  surface,  or  ap- 
proximately, i'  lat.  on  chart:  i'  long,  on  chart  =  i'  lat.  on 
spheroid:  i'  long,  on  spheroid.  If  this  relation  is  preserved,  it 
will  be  found  that  any  line  of  constant  bearing  (loxodrome  or 
rhumb  line)  will  be  represented  by  a  straight  line  on  the  chart. 


MERCATOR'S  PROJECTION 


275 


In  Fig.  107  let  AB  on  the  earth's  surface  be  represented  by 
A'E'  on  the  chart  (actual  size).  In  order  that  the  two  lines 
may  have  the  same  bearing  it  is  necessary  that 


or 


__  = _Rmd<j> 

dx      CB      Rpd\ 


dy 


dx 
Rpd\ 


Rmd<j>. 


(a) 


Pole 


CHART 


FIG.  107. 


In  other  words,  since  the  longitude  has  been  expanded  (in  the 

ratio  — -J  by  the  method  of  constructing  the  chart,  it  is  neces- 
sary to  expand  the  latitudes  in  the  same  ratio  in  order  to  preserve 
the  scale  and  give  AB  the  same  bearing.  Now  since  dx  is  rep- 
resented as  large  as  the  corresponding  arc  on  the  equator,  we 
have 

dx         ad\        a_ 
Rpd\~  Rpd\~  Rp 

Substituting  in  (a),  we  obtain 


276  MAP  PROJECTIONS 

or,  since  Rp  =  N  cos  <£ 

dy  =  1^L-.ad<t> 
Ncos<l> 


cos  0  (;  —  e2  sin2  <£) 


• 

cos</>(i  — 


Multiplying  e2  by  sin2  0  +  cos2  0,  the  integral  may  be  sep- 
arated into  two,  giving,  after  multiplying  numerator  and  de- 
nominator by  cos  <£, 

rcos  <ft  d<j>     _       C+   ecos<j>d(t> 
cos2  0  JQ    i  —  e2  sin2  0 


i  —  sin  0      2          i  —  e  sm  0 

where  M  —  0.4342945,  the  modulus  of  the  common  logarithms. 
Employing  the  formulae, 


,  i  +  sin  x 

and  — ! — ; —  =  tan 


the  equation  may  be  expressed 

«  sin 


r       i 
[133] 


in  which  y  is  in  the  same  linear  units  as  a. 

In  order  to  express  y  in  nautical  miles  or  minutes  of  arc  on  the 

equator  *  it  is  necessary  to  multiply  by  -  ,  giving, 

air 

*  The  Nautical  Mile  contains  6080.20  ft.;  this  is  not  identical  with  the  number 
of  feet  in  one  minute  of  arc  on  the  earth's  equator.  For  a  discussion  of  this  matter, 
see  Appendix  12,  Coast  Survey  Report  for  1881. 


MERCATOR'S  PROJECTION 


- 


277 


,      [134] 


or 

y 


=  7915.705  log  tan  U5°H  —  J  -  22^.945  sin  #-0.051  sin3<£.      [135] 


ico- 


120 


120i  100J  80" 

FIG.  108.    Mercator  Chart. 

Also  x  =  60  X  X°,  [136] 

the  unit  being  the  nautical  mile.    Values  of  y,  called  meridional 
parts,  will  be  found  in  works  on  navigation. 

This  chart  is  much  used  by  navigators  because  it  possesses 
the  property  that  the  bearing  of  any  point  B  from  a  point  A  as 


278  MAP  PROJECTIONS 

measured  on  the  chart  is  the  same  as  that  bearing  on  which  a 
vessel  must  sail  continuously  to  go  from  A  to  B.  The  track 
cuts  all  meridians  on  the  globe  at  the  same  angle,  just  as  a 
straight  line  on  the  chart  cuts  all  meridians  at  the  same  angle. 
This  track  is  not  the  shortest  one  between  A  and  B,  but  for 
ordinary  distances  the  length  differs  but  little  from  that  of  the 
great-circle  track.  In  following  a  great-circle  track  the  navi- 
gator transfers  to  the  Mercator  chart  a  few  points  on  the  great- 
circle  obtained  from  his  great-circle  chart,  by  means  of  their 
latitudes  and  longitudes  and  then  sails  on  the  rhumb  lines 
between  consecutive  plotted  points.  Fig.  108  shows  a  Mercator 
chart. 

183.  Rectangular  Spherical  Coordinates. 

A  system  of  rectangular  spherical  coordinates,  used  in  Europe, 
consists  in  referring  all  points  to  two  great  circles  through  some 
selected  origin,  one  of  them  being  the  meridian,  the  other  the 
prime  vertical.  Within  small  areas  these  coordinates  are  prac- 
tically the  same  as  rectangular  plane  coordinates.  When  the 
area  is  so  great  that  the  effect  of  curvature  becomes  appre- 
ciable, small  corrections  are  introduced,  so  that  the  form  of  the 
plane  coordinates  is  retained  without  loss  of  accuracy.  Such  a 
system  is  very  convenient  when  connecting  detail  surveys  with 
the  triangulation,  particularly  for  local  surveyors  who  may 
not  be  familiar  with  geodetic  methods  of  calculating  latitudes 
and  longitudes.  The  method  is  not  well  adapted  to  mapping 
very  large  areas.  (See  Crandall's  Geodesy,  p.  187.) 


CHAPTER  XII 

APPLICATION  OF  METHOD  OF  LEAST  SQUARES  TO 
THE  ADJUSTMENT  OF  TRIANGULATION 

184.  Errors  of  Observation. 

Whenever  an  observer  attempts  to  determine  the  values  of 
any  unknown  quantities,  he  at  once  discovers  a  limit  to  the 
precision  with  which  he  can  make  a  single  measurement.  In 
order  to  secure  greater  precision  in  his  final  result  than  can  be 
obtained  by  a  single  measurement,  he  resorts  to  the  expedient 
of  making  additional  measurements,  either  under  the  same  con- 
ditions or  under  different  conditions.  Under  these  circumstances 
it  will  be  observed  that  the  results  are  discordant  and  that  the 
same  numerical  result  almost  never  occurs  twice.*  The  ques- 
tion at  once  arises,  then,  What  are  the  best  values  of  the  un- 
known quantities  which  it  is  possible  to  obtain  from  these 
measurements  ? 

The  method  of  least  squares  has  for  its  main  objects  (i)  the 
determination  of  the  best  values  which  it  is  possible  to  obtain 
from  a  given  set  of  measurements,  and  (2)  the  determination 
of  the  degree  of  dependence  which  can  be  placed  upon  these 
values,  or,  in  other  words,  the  relative  worth  of  different  deter- 
minations; (3)  it  also  enables  us  to  trace  to  their  sources  the 
various  errors  affecting  the  measurements  and  consequently  to 
increase  the  accuracy  of  the  result  by  a  proper  modification  of 
the  methods  and  instruments  used.  The  method  is  founded 

*  This  is  only  true,  however,  when  the  observer  is  taking  each  reading  with  the 
utmost  possible  refinement.  If,  for  example,  angles  are  read  only  to  the  nearest 
degree,  the  result  will  always  be  the  same  no  matter  how  many  times  the  measure- 
ment may  be  repeated;  but  if  read  to  seconds  and  fractions,  they  will  in  general  all 
be  different. 

279 


28o  ADJUSTMENT  OF  TRIANGULATION 

upon  the  mathematical  theory  of  probability,  and  upon  the 
assumption  that  those  values  of  the  unknowns  which  are  ren- 
dered most  probable  are  the  best  that  can  be  obtained  from  the 
measurements. 

185.  Probability. 

If  an  event  can  happen  in  a  ways  and  fail  in  b  ways,  and  all 
of  these  ways  are  equally  likely  to  occur,  the  probability  that 
the  event  will  happen  in  any  one  trial  is  expressed  by  the  fraction 

— - — ,  and  the  probability  that  it  will  fail  is  expressed  by  — —  • 
a  +  b  a  -\-  b 

Since  it  must  either  happen  or  fail,  the  sum  of  the  two  prob- 
abilities represents  a  certainty.  This  sum  is  — —7  H —  =  i. 

a  +  b      a  +  b 

Therefore  the  probability  of  the  happening  of  an  event  is  repre- 
sented by  some  number  lying  between  o  and  i,  the  larger  the 
fraction  the  greater  the  probability  of  its  happening.  For  ex- 
ample, a  die  may  fall  so  that  any  one  of  its  six  faces  is  uppermost, 
and  all  of  these  six  possibilities  are  equally  likely  to  occur;  the 
probability  of  any  one  of  its  faces  being  up  is  £. 

186.  Compound  Events. 

If  a  certain  event  can  happen  in  a  ways  and  fail  in  b  ways, 
and  if  a  second,  independent,  event  can  happen  in  a'  ways  and 
fail  in  b'  ways,  and  all  are  equally  likely  to  occur,  then  the  total 
number  of  ways  in  which  the  events  can  take  place  together  is 
(a  +  b)  (af  +  b').  The  number  of  ways  in  which  both  can  hap- 
pen is  aa',  and  the  probability  of  its  happening  is  :— -  (  ,  ,.  • 

(a  +  b)  (a  -\-o ) 

For  example,  the  probability  of  double  six  being  thrown  with  a 
pair  of  dice  is  J  X  J  =  ^V  ^t  *s  evident  that  the  probability 
of  the  simultaneous  occurrence  of  two  events  is  the  product  of 
the  probabilities  of  the  occurrence  of  the  component  events. 
In  a  similar  way  it  may  be  shown  that  the  probability  of  the 
simultaneous  occurrence  of  any  number  of  independent  events 
is  the  product  of  their  separate  probabilities;  that  is,  if  PI,  PZ, 
P3  .  .  .  are  the  probabilities  of  the  occurrence  of  any  number 


COMPARISON  OF  ERRORS  281 

of  independent  events,  the  probability  of  their  simultaneous 
occurrence  is 

P  =  p,  x  P2  x  Pz  .  .  .  ,  [137] 

187.  Errors  of  Measurement  —  Classes  of  Errors. 

Every  measurement  of  a  quantity  is  subject  to  error,  of  which 
the  following  kinds  may  be  distinguished. 

1.  Constant  Errors. 

2.  Systematic  Errors. 

3.  Accidental  Errors. 

188.  Constant  Errors. 

A  constant  error  has  the  same  effect  upon  all  observations  in 
the  same  series  of  measurements.  For  instance,  if  a  steel  tape 
is  o.oi  ft.  too  long,  this  error  affects  every  100  ft.  measurement 
in  just  the  same  way. 

189.  Systematic  Errors. 

A  systematic  error  is  one  of  which  the  algebraic  sign  and  the 
magnitude  bear  a  fixed  relation  to  some  condition.  For  ex- 
ample, if  the  measurements  with  the  tape  are  made  at  different 
temperatures,  the  error  resulting  from  this  variation  of  tem- 
perature is  systematic  and  may  be  computed  if  the  tempera- 
tures and  the  coefficient  of  expansion  are  known. 

190.  Accidental  Errors. 

Accidental  errors  are  not  constant  from  observation  to  ob- 
servation; they  are  just  as  likely  to  be  positive  as  negative;  in 
general  they  follow  the  exponential  law  of  error,  as  will  be 
explained  later  (Art.  197).  The  error  of  placing  a  mark  opposite 
to  the  end  graduation  of  the  tape  is  of  this  class. 

191.  Comparison  of  Errors. 

There  is  in  reality  no  fixed  boundary  between  the  accidental 
and  the  systematic  errors.  Every  accidental  error  has  some 
cause,  and  if  the  cause  were  perfectly  understood  and  the  amount 
and  sign  could  be  determined,  it  would  cease  to  be  an  accidental 
error,  but  would  be  classed  as  systematic.  On  the  other  hand, 
errors  which  are  either  constant  or  systematic  may  be  brought 


282  ADJUSTMENT  OF  TRIANGULATION 

into  the  accidental  class,  or  at  least  made  to  partially  obey  the 
law  of  accidental  error,  by  so  varying  the  conditions,  instru- 
ments, etc.,  that  the  sign  of  the  error  is  frequently  reversed. 
If  a  tape  has  o.oi  ft.  uncertainty  in  length,  this  produces  a 
constant  error,  in  the  result  of  a  measurement.  If,  however,  we 
use  several  different  tapes,  each  with  an  uncertainty  of  o.oi  ft., 
this  error  may  be  positive  or  negative  in  any  one  case.  In  the 
long  run  these  different  errors  tend  to  compensate  each  other 
like  accidental  errors. 

In  the  class  of  systematic  errors  would  be  placed  such  errors 
as  those  due  to  changes  in  temperature,  light,  and  moisture, 
or  change  in  the  adjustments  of  instruments.  These  errors 
may  be  computed  and  allowed  for  as  soon  as  we  know  the  law 
governing  their  action,  or  they  may  be  partially  eliminated  by 
varying  conditions  under  which  the  measurements  are  made. 

Under  the  constant  class  comes  the  observer's  error,  which 
tends  to  become  constant  with  increased  experience  in  observing. 
This  error  may  be  allowed  for  as  soon  as  its  magnitude  and  sign 
have  been  determined,  or  it  may  be  eliminated  by  the  method 
of  observation.  Certain  errors  in  the  instrument  may  have  a 
constant  effect  on  the  result;  these  may  be  dealt  with  in  the 
same  manner  as  the  personal  error.  It  should  be  noticed  that 
after  the  constant  error  or  the  systematic  error  has  been  elimi- 
nated, there  still  remains  a  small  error  due  to  the  fact  that  the 
magnitude  of  the  constant  error  itself  was  not  perfectly  deter- 
mined or  that  its  elimination  was  imperfect.  This  remaining 
error  must  be  regarded  as  an  error  of  the  accidental  class,  since 
its  magnitude  is  unknown  and  it  is  just  as  likely  to  be  positive 
as  negative. 

Under  accidental  errors  are  included  all  those  which  are  sup- 
posed to  be  small  and  just  as  likely  to  be  positive  as  negative. 
They  are  due  to  numerous  unknown  causes,  each  error  being  in 
reality  the  algebraic  sum  of  many  smaller  errors.  Under  this 
class  may  be  noted  errors  in  pointing  with  a  telescope,  errors  in 
reading  scales  and  estimating  fractions  of  scale  divisions,  and 


ADJUSTMENTS  OF  OBSERVATIONS  283 

undetected  variations  in  all  of  the  conditions  governing  syste- 
matic errors. 

192.  Mistakes. 

These  are  not  errors,  but  they  must  be  considered  in  connec- 
tion with  the  discussion  of  accuracy  of  observations.  They  in- 
clude such  cases  as  reading  one  figure  for  another,  as  a  6  for  a  o, 
or  reading  a  scale  in  the  wrong  direction,  as  reading  46°  for  34°. 

193.  Adjustment  of  Observations. 

When  the  number  of  measurements  is  just  sufficient  to  de- 
termine the  quantities  desired,  then  there  is  but  one  possible 
solution,  and  the  results  must  be  accepted  as  the  true  values. 
When  additional  measurements  are  made  for  the  purpose  of 
increasing  the  accuracy  of  the  results,  this  gives  rise  to  discrep- 
ancies among  the  different  measurements  of  the  same  quantities, 
since  each  is  subject  to  errors.  The  method  of  least  squares 
enables  us  to  compute  those  values  which  are  rendered  most 
probable  by  the  existence  of  the  observations  and  in  view  of 
the  discrepancies  noted;  it  cannot,  however,  tell  us  anything 
about  the  existence  of  constant  errors,  unless  new  observations 
made  under  different  conditions  reveal  new  discrepancies.  For 
example,  if  a  pendulum  is  swung  and  certain  small  variations 
in  the  last  decimal  place  of  the  period  are  noticed,  these  may  be 
regarded  as  due  to  small  errors  in  the  running  of  the  chronom- 
eter and  to  accidental  errors  of  observing;  but  if  the  pendulum 
case  be  mounted  on  a  support  whose  flexibility  is  very  much 
greater  than  that  of  the  first,  and  larger  variations  are  now 
observed,  it  becomes  apparent  that  an  error  of  the  systematic 
class  is  affecting  all  our  observations,  though  it  does  not  appear 
at  all  in  the  first  observations,  because  all  the  measurements 
were  affected  alike.  An  investigation  of  the  law  governing 
this  error,  and  the  determination  of  its  magnitude  and  sign, 
enable  us  to  correct  the  result  for  such  part  of  the  error  as  we 
are  able  to  determine.  There  remains  in  the  result,  however, 
an  accidental  error,  namely,  the  error  in  the  measurement  of 
the  flexure  correction. 


284  ADJUSTMENT  OF  TRIANGULATION 

194.  Arithmetical  Mean. 

The  formulae  employed  in  adjusting  observations  are  usually 
made  to  depend  upon  the  axiom  that  if  a  number  of  observations 
be  made  directly  upon  the  same  quantity,  all  made  under  the 
same  conditions  and  with  the  same  care,  the  most  probable 
value  of  the  quantity  sought  is  the  arithmetical  mean  of  all 
the  separate  results;  that  is,  if  the  results  of  the  observations 
are  MI,  Mz,  Mz,  .  .  .  Mn,  the  most  probable  value  of  the  quan- 
tity, MQ,  is  given  by 


[I38] 

It  is  to  be  carefully  noted  that  this  is  not  the  true  value,  M, 
but  simply  the  most  probable  value  under  the  circumstances; 
if  additional  measurements  be  made,  MO  changes  correspond- 
ingly in  value,  because  we  know  more  about  its  real  value  than 
we  did  at  first. 

195.  Errors  and  Residuals. 

It  now  becomes  necessary  to  distinguish  between  errors  and 
residuals.  The  error  is  the  difference  between  any  measured 
value  and  the  true  value.  Its  magnitude  can  never  be  known, 
because  the  true  value  can  never  be  known.  The  residual  is 
the  difference  between  a  measured  value  and  the  most  probable 
value.  This  is  a  quantity  which  may  be  computed  for  any  set 
of  observations.  In  a  set  of  very  accurate  observations  which 
are  free  from  constant  and  systematic  errors  the  residual  is  a 
close  approximation  to  the  true  error.  It  may  be  shown  that 
for  the  case  of  direct  observations  the  algebraic  sum  of  the 
residuals  is  zero;  that  is,  if  we  compute  Vi  =  Mi  —  M0,  ih  = 
M2  —  M0,  etc.,  then  ^v  =  o,  where  Vi,  %  .  .  .  are  the  residuals. 

196.  Weights. 

In  case  the  measurements  are  of  different  degrees  of  relia- 
bility, they  are  given  different  weights.  The  weight  of  an 
observation  may  be  regarded  as  the  number  of  times  the  ob- 
servation is  repeated  and  the  same  numerical  result  obtained. 


DISTRIBUTION  OF  ACCIDENTAL  ERRORS 


285 


It  expresses  the  relative  worth  of  different  measured  values. 
Weights  are  purely  relative  and  may  be  computed  on  any  base 
desired.  To  say  that  two  measurements  have  weights  2  and  i 
respectively,  is  the  same  as  saying  that  they  have  weights  \ 
and  \.  '  From  the  above  definition  it  is  apparent  that  the  weighted 
mean  is  expressed  by 


that  is,  the  weighted  mean  is  found  by  multiplying  each  ob- 
servation by  its  weight,  adding  the  results,  and  dividing  by  the 
sum  of  the  weights. 

Multiplying  an  observation  (Mi)  by  its  weight  (pi)  is  the 
same  as  taking  pi  observations  each  equal  in  value  to  MI. 


197.  Distribution  of  Accidental  Errors. 

An  inspection  of  the  results  of  a  large  number  of  measure- 
ments will  show  that 

(1)  H-  and  —  errors  are  equally  numerous. 

(2)  Small  errors  are  much  more  numerous  than  large  ones. 

(3)  Very  large  errors  seldom  occur. 

The  curve  which  expresses  the  law  of  variation  of  such  errors 
will  be  of  the  form  shown  in  Fig.  109.     In  accordance  with  (i) 


286 


ADJUSTMENT  OF  TRIANGULATION 


the  curve  is  symmetrical;  in  accordance  with  (2)  its  maximum 
is  at  the  axis  of  F;  from  (3)  it  is  evident  that  the  curve  cuts 
the  axis  of  X  at  some  distance  from  0. 

The  manner  in  which  observations  are  affected  by  accidental 
errors  is  shown  by  the  "shot  apparatus  "  shown  in  Fig.  no.    A 


FIG.  no.     "Shot  Apparatus." 

large  number  of  small  shot,  representing  observations,  are 
allowed  to  drop  through  an  opening  in  the  middle  of  the  case. 
If  there  were  no  obstructions  the  shot  would  fall  directly  into 
the  central  (vertical)  compartment.  Between  the  opening  and 
the  vertical  compartments  a  number  of  pegs  are  interposed, 
each  representing  a  source  of  error  or  deflection  of  the  shot 
from  its  natural  course.  vThe  shot  are  therefore  diverted  some- 


DISTRIBUTION  OF  ACCIDENTAL  ERRORS  287 

what  from  a  straight  course  and  arrange  themselves  in  the 
different  columns  in  the  manner  shown.  The  curve  joining  the 
tops  of  the  columns  is  seen  to  resemble  closely  the  "curve  of 
error." 

In  order  to  obtain  a  formula  expressing  the  law  of  error  we 
suppose  the  curve  asymptotic  to  the  axis  of  X,  and  write  the 
equation  of  the  curve  in  the  general  form 

y=f(x),  [140] 

where  x  represents  the  magnitude  of  an  error  and  y  the  fre- 
quency with  which  this  error  occurs  on  a  large  number  of  measure- 
ments; /  represents  some  unknown  function  of  x.  It  is  neces- 
sary to  assume  that  the  number  of  observations  is  very  large; 
otherwise  the  supposed  balancing  of  +  and  —  errors  will  be 
imperfect.  The  true  error  x  can  never  be  known,  but  the 
distribution  of  the  residuals  about  the  most  probable  value 
will  evidently  follow  the  same  general  law,  so  we  may  write 
also 

y=f(v)  [141] 

as  the  law  to  which  the  residuals  must  conform.  This  equation 
also  expresses  the  probability  of  the  occurrence  of  a  residual  v. 

If  we  let  the  total  area  between  the  curve  and  the  axis  of  X  be 
represented  by  unity,  then  the  probability  that  a  certain  residual 
will  fall  between  the  limits  v  and  v  +  dv  will  be  represented  by 
the  area  included  between  the  curve,  the  X  axis,  and  the  two 
ordinates  at  v  and  v  +  dv,  since  in  the  long  run  the  number  in  a 
given  column  will  be  proportional  to  the  probability  expressed 
by  the  ordinate  at  that  point,  that  is, 

ydv  =  f  (v)  dv.  [142] 

If  we  suppose  n  observations  of  equal  weight,  giving  the  results 
MI,  M2,  .  .  .  Mn,  to  be  made  on  any  functions  of  the  unknowns 
Zi,  zi,  .  .  .  ,  z»,  giving  the  residuals  vi,  %,  .  .  .  ,  vn,  then  the 
probability  of  the  occurrence  of  these  residuals  is  /  (v\)  dv, 
dv  .  .  .  f  (vn)  dv.  The  probability  of  the  simultaneous 


288 


ADJUSTMENT  OF  TRIANGULATION 


occurrence  of  these  residuals  is  the  product  of  the  separate 
probabilities,  that  is, 

P  =  /  Oi)  do  X  /  (%)  dv  X   .  .  .  /  (vn)  <fo.  [143] 

or,  taking  logs  of  both  members  of  the  equation, 
logP  =  log/Oi)  +  log/ 02)  +  •  •  •  log/ (O  +  n  X  log  do. 

The  results  desired  for  21,  22,  etc.,  are  those  for  which  the  prob- 
ability of  the  occurrence  of  Vi,  %,  .  .  .  is  a  maximum.  Therefore 
P  must  be  a  maximum.  To  find  the  conditions  for  this  maxi- 
mum, differentiate  log  P  with  respect  to  each  variable,  Zi,  22,  .  .  , 
and  place  the  results  equal  to  zero.  This  gives 

a  log  P  ^     i       d/  pi)  i     _  df  On) 

d*i         /OO  "    dsi  /On)  "    d«i 

alogP       i     sfM  .  i 


=  o. 


+ 


044] 


But  we  observe  that 


in  which  /'  represents  some  new  function  of  v. 


=F(v), 


For  brevity  place  -f 

tw 

Then  Equa.  [144]  become 


These  equations  contain  all  the  unknown  quantities  (2)  and 
are  equal  in  number  to  the  number  of  unknown  quantities. 


DISTRIBUTION  OF  ACCIDENTAL  ERRORS  289* 

Hence,  if  the  form  of  the  function  F  were  known,  the  solution  of 
these  equations  would  give  the  most  probable  values  of  Zi,  22,  etc. 

The  above  equations,  being  perfectly  general,  hold  true  for  all 
cases,  so  they  must  hold  true  for.  any  special  case.  The  form  of 
F  determined  for  the  special  case  must  therefore  be  the  form  of 
this  function  for  all  cases. 

Consider  n  direct  observations  of  equal  weight  on  one  unknown 
quantity  Zi,  the  results  of  the  measurements  being  MI,  Mi, 
.  .  .  Mn,  and  the  residuals  being  denoted  by  Vi,  %,  .  .  .  vn. 
The  most  probable  value  of  z\  is  given  by 

Zi  =  MI  —  Vi  =  M2  —  ih  =   .  .  .  Mn  —  vn. 

Differentiating  with  respect  to  Zi, 


Substituting  these  values  in  Equa.  [147],  we  obtain 

F  fa)  +  F  (%)  +  -  •  •  +  F  (vn)  =  o.  (b) 

But  in  this  special  case  (Art.  195), 

*i  +  %  4-  •  •  •  +  vn  =  o.  (c) 

Hence,  if  both  Equa.  (b)  and  (c)  are  true.  F  must  signify  mul- 
tiplication by  a  constant;  that  is, 

F  0)  =  cv.  [148] 

Substituting  in  Equa.  [146]  and  [145], 
df  (v)      f,\       dv 

*  -/w-"5- 

and  77-,'*®-  «?•' 

/  (v)       dz  dz 

Integrating  both  members, 

log/  W  =  i  CT2  4-  c'. 
Therefore  /  (v)  = 


Substituting  this  in  the  equation  of  the  curve  of  error  (y=f  (v)), 
we  have 

y  = 


290  ADJUSTMENT  OF  TRIANGULATION 

In  reality  y  decreases  as  v  increases ;  the  exponent  of  e  is  there- 
fore negative,  and,  since  the  constants  may  be  combined,  we  have 

y  •=  ke~h2*,  [149] 

in  which  h2  and  k  are  constants  depending  upon  the  character  of 
the  observations.  This  equation  expresses  the  law  in  accordance 
with  which  the  residuals  must  be  distributed  in  order  to  give  a 
maximum  value  of  P.  If  we  replace  v  by  x,  the  equation  also 
shows  the  law  governing  the  distribution  of  the  actual  errors. 

It  is  important  to  note  that  the  law  governing  the  distribution 
of  accidental  errors  holds  true  in  the  long  run;  in  order  to  have  a 
close  agreement  of  the  theory  with  the  results  actually  observed 
it  is  essential  that  the  number  of  observations  should  be  very 
large.  With  a  limited  number  of  observations  we  should  expect 
that  the  residuals  would  follow  the  law  only  approximately. 

198.   Computation  of  Most  Probable  Value. 

From  Equa.  [143]  we  have  seen  that 
P  =  /  W  X  /  (%)  .  .  .   X  /  (fln)  (dv)n  -  a  maximum.      [150] 

Applying  Equa.  [149],  this  becomes 

±   =  Knc,       i     z     '  -  •  n>  ((ii)\n  =  a,  maximum.  1^1 

V         /  L       J       J 

It  is  evident  that  P  is  a  maximum  when  ' 

v\   +  %2  +  •  •  •  ^n2  =  a  minimum,  [152] 

that  is,  when  the  sum  of  the  squares  of  the  residuals  has  its  least 
value. 

Equa.  [147]  express  the  conditions  necessary  to  make  P  a 
maximum  or  to  make  the  sum  of  the  squares  of  the  residuals  a 
minimum.  Since  the  function  F  means  multiplication  by  a  con- 
stant, Equa.  [147]  become 

£? .  o. 

dvn  _ 

[lS3l 

fan 

Cj%tft 

w 


RELATION  BETWEEN  H  AND  P 


291 


These  equations  are  equal  in  number  to  the  number,  q,  of  un- 
known quantities,  and  their  simultaneous  solution  gives  the  most 
probable  values  of  the  unknown  quantities.  They  are  usually 
called  Normal  Equations. 

199.  Weighted  Observations. 

If  the  observations  are  of  different  weights,  each  observation 
equation  should  be  used  (Art.  196)  the  number  of  times  denoted 
by  its  weight.  Hence,  in  forming  the  normal  equations  we 
should  multiply  each  observation  equation  by  the  coefficient  of 
the  unknown  and  by  the  weight  of  the  equation.  The  normal 
equations  in  this  case  are  as  follows: 


=  o. 


=  o. 


=  o. 


[154] 


This  same  result  will  be  obtained  if  we  first  multiply  each  ob- 
servation equation  by  the  square  root  of  its  weight.  This  shows 
that  multiplying  a  set  of  equations  by  the  square  roots  of  their 
weights  reduces  them  all  to  observations  of  weight  unity  (equal 
weights). 

200.  Relation  between  h  and  p. 

If  the  n  observations  have  weights  pi,  pz,  .  .  .  ,  and  the  con- 
stant h  is  hi,  fa,  .  .  .  for  these  observations,  then 

p  =  feer-W  .  fee-*"*  .  .  . 
=  fefe  .  .  .  £n 


and  /*iW  +  feW  +  •  •  •  is  to  be  a  minimum.  [156] 

The  conditions  for  this  minimum  are 


dl' 


o. 


=0. 


[157] 


2Q2  ADJUSTMENT  OF  TRIANGULATION 

Equas.  [154]  and  [157]  express  the  same  conditions. 
Hence  pi  '•  pz  '•  •  •  •   —  h\   :  h?  :  .  .  .  , 

showing  that  the  weight  of  an  observation  varies  as  the  square  of 
the  constant  h  for  the  observation.  Consequently  the  more 
accurate  the  observation  the  greater  the  value  of  h. 

Example.  As  an  illustration  of  the  manner  of  applying  these  equations  to  the 
computation  of  the  most  probable  values  of  the  unknowns,  suppose  that  at  a  tri- 
angulation  station  0  (Fig.  in),  the  angles  have  been  measured  as  shown. 

Denoting  the  most  probable  values  of  these 
angles  by  0i,  02,  and  z3,  the  measurements  are 
given  by  the  following  equations: 


02  = 

03  = 

Zl  +  22  = 

Z2  +  Z3  = 

Z2  +  Z3  = 


31  10  i.o, 

40  50  10    .0, 

42  10  IQ    .7, 

72  OO  26    .O, 

4  10  46    .0, 

83  00  30    .2. 


Denoting  by  vi,  vz,  etc.,  the  residuals  of  the 
different  measurements,  these  may  be  written 


02  —  40 

z3  -  42 

zi  +  z2  —  72 

+  z2  +  03  —  114 

Z2  +  Z3-      83 


10  17  .o  =  DI, 

5O  IO  .0  =  1)2, 

10  19  .7  =  VS, 

OO  26  .O  =  »4, 

10  46  .0  =  1)5, 

00  30  .2  =  Z>6, 


n.  which  are  called  observation  equations. 

If  we  apply  equations  (153),  differentiating 

each  v  with  respect  to  the  three  unknown  quantities  in  succession,  we  obtain  the 
normal  equations. 

3  Zl  +  2  02  +       Z3  —   217°  2l'  29".0   =  O, 
2  0i  +  4  02  +  2  03  —  310     01     52     .2   =  O, 

Zi  +  2  02  +  3  03  —  239   21  35  .9  =  o. 
Solving  these  simultaneously,  we  obtain 

01  =  31°  10'  i6".4S, 

02  =   40     50    09    -875, 

z3  =  42    10  19  .90. 


These  are  the  most  probable  values  of  the  angles. 


SOLUTION  BY  MEANS  OF  CORRECTIONS  293 

201.  Formation  of  the  Normal  Equations. 

It  should  be  observed  that  since  the  observation  equations  are 
linear  in  this  case,  the  differential  coefficients  are  equal  to  the 
numerical  coefficients.  Hence,  to  form  the  normal  equations  we 
may  proceed  as  follows :  For  each  unknown,  form  a  normal  equation 
by  multiplying  each  observation  equation  by  the  numerical  coefficient 
of  the  unknown  in  that  equation,  adding  these  results  and  placing  the 
sum  equal  to  zero.  This  rule  is  simply  a  statement  in  words  of 
what  is  expressed  in  Formula  [153]  as  applied  to  linear  equations. 
If  the  observations  are  of  different  weights,  the  only  change  in 
the  above  rule  is  that  each  observation  equation  is  multiplied 
by  its  weight  as  well  as  by  the  coefficient  of  the  unknown. 

In  regard  to  the  observation  equations  it  should  be  understood 
that  they  are  not  like  ordinary  equations.  They  are  often 
written,  however,  with  zero  hi  place  of  the  v  in  the  right  hand 
member.  Observation  equations  cannot  be  multiplied  by  any 
number  or  combined  with  each  other  (except  when  forming  nor- 
mal equations) ;  for  if  this  is  done,  the  weight  of  the  observation  is 
thereby  changed. 

202.  Solution  by  Means  of  Corrections. 

If  the  independent  terms  *  in  the  observation  equations  are 
large,  it  will  often  save  labor  in  the  calculations  if  we  place  the 
unknown  quantity  Zi  equal  to  an  approximate  value  M\  plus  a 
correction  Zi,  Z2  =  M2  -f  22,  etc.  Substituting  these  values  in  the 
original  observation  equations,  we  obtain  a  new  set  of  equations 
in  terms  of  the  corrections  and  in  which  the  independent  terms 
will  be  small.  By  forming  normal  equations  and  solving  as  be- 
fore, we  find  the  most  probable  values  of  the  corrections.  Adding 
these  corrections  to  the  approximate  values,  we  find  the  most 
probable  values  of  the  unknown  quantities  themselves. 

*  The  independent  term  in  any  equation  is  that  term  which  does  not  contain 
any  of  the  unknowns. 


294  ADJUSTMENT  OF  TRIANGULATION 

Example.  In  the  example  just  solved,  suppose  we  assume  for  the  approximate 
values  the  results  of  the  direct  measurements,  and  let  zi,  z2,  etc.,  represent  the  most 
probable  corrections.  Then  the  observation  equations  become 

zi       =  o, 

z2       =  o, 

z3       =  o, 

Zl  +  Z2  +  l".O  =  O, 

zi  +  z2  +  z3  +  o  .7  =  0, 
z2  +  z3  —  o  .5  =  0. 
Forming  the  normal  equations  as  before,  we  have 

3  zi  +  2  z2  +     z3  +  i"-7  =  o, 

2  Zi  +  4  Z2  +  2  Z3  +   I     .2=0, 
Zl  +  2Z2  +  3Z3  +  O    .2   =  O. 

The  solution  of  these  equations  gives 

zi  =  -o".55, 
z2  =  — o  .125, 

Z3  =    +0    .20, 

which,  added  to  the  values  observed  directly,  give  the  same  results  as  before. 

203.   Conditioned  Observations. 

If  the  quantities  sought  are  not  independent  of  each  other,  but 
are  subject  to  certain  conditions,  the  solution  must  be  modified 
accordingly.  Each  observation  gives  rise  to  an  observation 
equation,  and  each  condition  may  be  expressed  by  a  condition 
equation.  The  solution  may  be  effected  by  eliminating,  between 
the  two  sets  of  equations,  as  many  unknowns  as  there  are  equa- 
tions of  condition.  From  the  remaining  equations  we  may  form 
the  normal  equations  and  solve  for  the  most  probable  values  of 
the  unknowns.  Substituting  these  values  back  in  the  original 
condition  equations,  we  obtain  the  remaining  unknowns. 

Example.  The  three  angles  of  a  triangle  are  A  =  61°  07'  52".oo,  B  =  76°  50'- 
54;'.oo,  and  C  =  42°  01'  i2".i5.  The  spherical  excess  is  02". n.  The  weights 
assigned  to  the  measured  angles  are  3,  2,  and  2,  respectively.  These  angles  are 
subject  to  the  fixed  relation  A  +  B  +  C  =  180°  oo'  02".! i. 

Letting  t»i,  v2,  t>3  be  the  most  probable  corrections  to  the  observed  values,  the 

observation  equations  are 

vi  =  vi,  wt.  3 

*  =  *,  "     2 

V3  =  V3,  "     2 

and  the  condition  equation  is 

vi  +  n  +  vs-  3"-96  =  o.  (<*) 


=  +o".9o, 


ADJUSTMENT  OF  TRIANGULATION  295 

Eliminating  v3,  there  remain 

*>i  =  PI,  wt.  3 

V2  =  V»,  "2 

*=-*-*  +  3"  96    "    2 
Forming  the  normal  equations  and  solving, 

*  =  +o".99, 

V2  =   +1    .485. 

Substituting  these  values  in  equation  (d), 

These  corrections,  added  to  the  measured  angles,  give  the  adjusted  angles,  as 
follows: 

A  =61°  07'  52".oo, 
B  =  76  50  55  -48, 
C  =  42  01  13  .64. 

Notice  that  the  discrepancy  is  distributed  inversely  as  the  weights.  This  will 
always  be  the  case  when  each  unknown  is  directly  observed,  and  there  is  but  one 
equation  of  condition;  that  is,  the  correction  to  the  first  is 

1,1,1  X  +3 
and  the  correction  to  the  second  is 

The  correction  to  the  third  is  the  same  as  the  correction  to  the  second. 

204.  Adjustment  of  Triangulation. 

The  adjustment  of  the  angles  of  a  triangulation  net  naturally 
divides  itself  into  two  parts:  (i)  the  adjustment  for  the  dis- 
crepancies arising  at  each  station,  and  (2)  the  adjustment  of 
the  figure  as  a  whole.  According  to  theory  these  should  all  be 
adjusted  simultaneously  in  order  to  obtain  the  most  probable 
values  of  the  angles.  The  usual  practice,  however,  is  to  deal 
with  the  two  separately.  The  local,  or  station,  adjustment  is 
made  first  if  the  method  of  observing  is  such  that  a  local  adjust- 
ment is  required.  If  the  observations  are  made  in  accordance 
with  the  program  given  in  Art.  44  (sec.  2,  Coast  Survey  in- 
structions), no  station  adjustment  is  necessary.  If  the  angles  are 
measured  by  the  repetition  method  and  the  horizon  is  closed,  the 
error  is  distributed  in  inverse  proportion  to  the  weights  (see 
Art.  203).  If  there  are  conditions  existing  among  the  angles, 


296  ADJUSTMENT  OF  TRIANGULATION 

due  to  measuring  sums  of  the  different  single  angles,  the  adjust- 
ment may  be  effected  by  expressing  these  as  condition  equations 
and  then  forming  normal  equations  and  solving,  as  in  the  ex- 
ample, p.  294. 

This  method  of  making  the  local  adjustment  first  is  justified, 
not  only  on  the  ground  of  saving  labor,  but  also  because  of  the 
well-known  fact  that  the  most  serious  errors  are  those  due  to 
eccentricity  of  signal  and  instrument,  phase  of  signal,  refraction, 
etc.,  which  do  not  appear  to  any  large  extent  in  the  local  ad- 
justment but  which  do  appear  in  the  figure  adjustment.  If 
we  compute  the  precision  of  angles  from  the  discrepancies  noted 
at  each  station,  and  then  estimate  from  these  values  the  error 
of  closure  to  be  expected  in  the  triangle,  we  find  that  these  are 
smaller  than  the  errors  of  closure  actually  occurring,  showing 
the  presence  of  constant  errors,  which  do  not  appear  in  the 
local  adjustment. 

205.  Conditions  in  a  Triangulation. 

The  geometric  conditions  connecting  the  angles  in  a  net  are 
of  two  classes:  (i)  those  which  express  the  relation  among  the 
angles  of  a  triangle  or  other  figure,  and  (2)  those  which  express 
the  relation  existing  among  the  sides  of  the  figure.  If  we  plot, 
for  example,  a  quadrilateral  figure,  starting  from  one  side  as 
fixed,  we  shall  find  that  if  the  sum  of  the  angles  in  three  of  the 
triangles  equals  their  theoretical  sums,  all  sums  in  the  other 
triangles  will  also  (necessarily)  equal  their  theoretical  amounts, 
namely,  180°  +  e" .  This  shows  that  of  all  the  possible  angle 
equations  which  might  be  written  for  this  figure  only  three  are 
really  independent. 

In  order  to  determine  the  number  of  angle  equations  in  any 
net,  let  s  be  the  total  number  of  stations,  su  the  number  of 
stations  not  occupied,  /  the  total  number  of  lines  in  the  figure, 
and  /i  the  number  of  lines  sighted  over  in  one  direction  only; 
then  the  number  of  angle  equations  in  the  figure  is 

+  s«  +  i.  [159] 


ADJUSTMENT  OF  A  QUADRILATERAL  297 

In  a  triangle  it  is  necessary  that  all  stations  should  be  occu- 
pied and  that  all  lines  should  be  sighted  over  in  both  directions, 
in  order  to  have  one  angle  equation,  that  is, 


If  a  new  station  is  added,  it  must  be  occupied  and  the  two  lines 
sighted  over  in  both  directions,  in  order  to  yield  a  new  angle 
equation.  If  this  is  done,  the  quantity  /  —  s  is  increased  by 
2  —  1  =  1.  If  a  line  is  drawn  between  two  stations  already 
located,  /  is  increased  by  i  and  there  is  a  new  angle  equation 
corresponding.  For  each  new  line  sighted  in  one  direction 
only,  /  is  increased  by  i  and  h  is  increased  by  i,  so  that  the  total 
is  unchanged. 

The  number  of  side  equations  in  a  net  may  be  estimated  as 
fellows:  Starting  with  one  line  as  fixed,  it  is  evidently  neces- 
sary to  have  two  more  sides  in  order  to  fix  a  third  point.  Hence, 
in  order  to  plot  a  figure,  we  must  have  at  least  2  (s  —  2)  lines 
hi  addition  to  the  base,  that  is,  2  5  —  3  lines  in  all.  Any  addi- 
tional lines  used  must  conform  to  those  already  used,  in  order 
to  give  a  perfect  figure;  hence  the  number  of  conditions  giving 
rise  to  side  equations  will  equal  the  number  of  superfluous  lines, 
that  is,  /  —  2  s  +  3,  where  /  is  the  total  number  of  lines  and  s 
is  the  number  of  sta'tions.  It  should  be  observed  that  while 
the  side  equation  is  primarily  a  relation  among  the  sides,  it  is 
also  a  relation  among  the  sines  of  the  angles,  and  this  fact  en- 
ables us  to  adjust  the  figure  by  altering  the  angles. 

206.  Adjustment  of  a  Quadrilateral. 

For  any  quadrilateral  figure  in  which  all  of  the  (eight)  angles 
have  been  measured  there  may  be  found  three  equations  which 
express  the  condition  that  the  triangles  must  all  "close."  There 
are  more  than  three  equations  which  may  be  formed;  but  if 
any  three  of  these  equations  are  satisfied,  the  others  necessarily 
follow  and  hence  are  not  independent.  There  will  also  be  one 
side  equation  expressing  the  condition  that  the  length  of  a  side 
(AB\  when  computed  from  the  opposite  side  (CD),  is  exactly 


298 


ADJUSTMENT  OF  TRIANGULATION 


the  same,  no  matter  which  pair  of  triangles  is  employed  in  the 
computation. 

In  selecting  the  three-angle  equations  we  may  take  any  three 
triangles  and  write  an  equation  for  each  expressing  the  con- 
dition that  the  sum  of  the  three  angles  equals  180°  +  e" .  It 
is  advantageous  in  this  case  to  avoid  triangles  having  small 
angles.  In  selecting  the  side  equation  it  is  well,  however,  to 
select  one  involving  small  angles,  so  as  to  give  large  coefficients 
of  the  corrections.  If  the  angle  equations  were,  also  chosen  so 
as  to  involve  the  small  angles,  the  solution  would  be  likely  to 
prove  unstable,  on  account  of  the  equality  of  some  of  the 
coefficients. 


B 


FIG.  112. 


FIG.  113. 


A  convenient  method  of  writing  a  side  equation  is  to  select 
some  point,  called  the  pole,  and  write  the  three  directions  from 
it  to  the  other  stations  in  the  order  of  azimuths.  For  example, 
taking  the  pole  at  A,  Fig.  112,  write  first 

AB-AD-AC. 
Then  from  this  write  the  ratios 

AB  AD  AC 

AD'AC'AB' 

the  method  of  forming  which  is  evident.    If  we  now  replace 


ADJUSTMENT  OF  A  QUADRILATERAL  299 

each  line  by  the  sine  of  the  angle  opposite  to  it  in  the  triangle 
which  is  indicated  by  the  fraction,  and  place  the  whole  equal 
to  unity,  we  have 

smADB      sinACD      sin  ABC  _  .      , 

sinABD      smADC      smACB 

It  may  be  shown,  by  solving  the  different  triangles  and  elimi- 
nating the  sides,  that  this  equation  expresses  the  condition  that 
the  length  of  AB  as  computed  from  CD  is  the  same  no  matter 
which  route  is  followed  in  the  computation. 

Problem.  Prove  by  a  direct  solution  of  the  triangles  in  Fig.  112  that  Equation 
[160]  is  true. 

Designating  the  angles  by  means  of  the  numbers  shown  in 
Fig.  113,  the  equation  becomes 

sin  2  sin  (4  +  5)  sin  8 

—  —  7  -  TT  —  ;  -  ;  -  =  I. 

sin  (i  -f  8)  sin  3  sin  5 

Before  this  equation  can  be  used,  however,  it  is  practically 
necessary  to  reduce  it  to  linear  form,  since  an  application  of 
Equa.  [153]  to  any  but  linear  equations  would  be  complicated. 

Suppose  our  equation  to  be  put  in  the  general  form 

sin  (Mi  +  vi)      sin  (M3  +  q,)  f      , 

' 


in  which  the  angle  is  written  as  an  approximate  value  M  plus  a 
small  correction  v.  Taking  logs  of  both  members  and  then 
applying  Taylor's  theorem,  we  have,  neglecting  squares  and 
higher  powers, 

log  sin  Mi  +  —  (log  sinMi)  i*  +  •  •  • 
oJVL\ 

-  (logsinM2  +  —  (logsinM2)  fy  +  •  •  •  j  =  o.     [163] 
The  quantity  -—  (log  sin  MI)  is  the  variation  per  i"  in  a 


300  ADJUSTMENT  OF  TRIANGULATION 

table  of  log  sines,  the  correction  v  being  in  seconds.     Hence, 
placing  61  =  —  —  -  (log  sin  Mi),  etc.,  we  have 


-  to  +  to  -  to  +  •  •  • 

+  log  sin  Mi  -  log  sin  M2  +  •  •  •  =  o.         [164] 

The  algebraic  sum  of  the  log  sines  represents  the  amount  by 
which  they  fail  to  satisfy  the  condition  equation.  Placing  this 
sum  equal  to  /,  the  side  equation  given  above  becomes 

to  +  54+5^4+5  +  to  —   (8l+8&l+8  +  to  +  to)   —1  =  0.         [165] 

Example.    Let  us  suppose  that  the  measured  angles  are  (Fig.  113), 

1.  6i°o7'52".oo 

2.  38  28  34  .90 

3.  38  22  19  .10 

4.  42  or  12  .15 

5.  29  14  32  .85 

6.  70  21  59  .20 

7.  49  26  21  .85 

8.  30  57  °7  -io 

These  angles  are  supposed  to  have  been  adjusted  for  local  conditions. 

To  form  the  angle  equations,  take  the  triangles  ABD,  ADC,  and  ABC  for  which 
the  values  of  the  spherical  excess  are  i".36,  i".77  and  i".o2,  respectively.  The 
computation  is  shown  in  tabular  form  as  follows: 

1  +  8  92°04'59".io 
2  38  28  34  .90 
7  49  26  21  .85 

179  59  55   -85 

180  oo  01   .36 


3  38°  22'  r9".io 
4  +  5  71  15  45  .00 

6  70  21  59  .20 
1  80  oo  03  .30 
180  oo  01  .77 


5  29°  14'  32".85 

6  70  21  59  .20 

7  49  26  21  .85 

8  30  57  07  .10 
180  oo  01  .00 
180  oo  01  .02 

+0".02 


ADJUSTMENT  OF  A  QUADRILATERAL 


This  gives  for  the  three  angle  equations 

(i  +  8)  +  2  +  7  =  180°  oo'  oi".36, 
3  +  (4  +  S)  +  6  =  180  oo  01  .77, 
5  +  6  +  7  +  8  =  180  oo  01  .02, 

or,  written  as  corrections, 

fi-B  +  %  +  V7  -  5.51        =  o, 

t>3  +  f 4+5  +  V6 +1-53  =0, 

V  6  +  V  6  +  V  7  +  »8  —  O.O2    =  O. 

To  form  the  side  equation,  take  the  pole  at  A.    Then  we  have 


giving 


AB    AD    AC_ 
AD'  AC'  AB' 

sin  (4  +  5)    sin  8 
sin  5 


sin  (i  +  8)         sin  3 


or 


log  sin  2+log  sin  (4+5)+log  sin  8— log  sin  (1+8)— log  sin  3— log  sin  5=0. 


The  computation  of  the  constant  term  of  this  equation  is  given  in  the  following 
table.  The  log  sines  of  those  angles  appearing  in  the  numerator,  together  with 
their  diff.  for  i"  (in  units  of  the  6th  place  of  decimals)  are  placed  in  the  left-hand 
column,  and  those  in  the  denominator  are  placed  in  the  right-hand  column.  The 
constant  /  is  the  difference  in  the  sums  of  the  log  sines. 


Angle. 

log  sine  (+). 

Diff.  i". 

Angle. 

log  sine  (—  ). 

Diff.  i". 

2 

4+5 
8 

9.7939242 
9.9763501 
9.7112329 

+  2.65 
+0.72 
+3-51 

1+8 
3 
5 

9.9997129 
9.7929268 
9.6888702 

-0.08 
+  2.66 
+3.76 

9.4815072 

9.4815099 

72 

-27 

Therefore  /  =  -2.7. 

The  side  equation  becomes 

2.65  %  +  0.72  t;4+s  +  3.51  t>8  +  0.08  fli+8  —  2.66  v3  —  3.76 1>6  -  2.7  =  o. 
Since  the  observations  are  direct,  all  of  the  observation  equations  take  the  form 


The  eight  observation  equations  and  the  four  condition  equations  are  now 
written,  and  we  are  ready  to  adjust  the  quadrilateral. 


302 


ADJUSTMENT  OF  TRIANGULATION 


207.   Solution  by  Direct  Elimination. 

If  we  select  for  the  four  independent  unknowns  ?;2,  %,  %,  and 
%,  and  express  the  four  conditions  in  terms  of  these,  we  have 

3-OI5^  -  5.282%  -4.751%  +  6.609%  +  0.2351, 


+4.282%  +  5.751%  -   5.609%  +3.725, 


%        =    +4.015%  -   5.282%  -   5.751%  +  5.609%  -   5.255, 

v7     =  -4-015  %  +  5-282  **  +  4-751  v$  -  6.609  ^8  +  5.275. 
%     =  .     V*. 

From  these  we  form  the  following  normal  equations  (Art. 
198,  Equa.  [153]): 


111. 

•b 

»»• 

ft 

Const. 

+58.450 
-75.534 
-79-581 
+91.501 

-  75-534 

+  IO2  .036 
+  105.193 
-123.474 

-    79.581 
+  105.193 
+  112.292 
-127.388 

+    9I-50I 
—  123.474 
-127.388 
+  151.282 

-56.527 
+  70.329 
+  75.589 
-83-678 

The  simultaneous  solution  of  these  equations  will  give  the 
most  probable  values  of  the  corrections. 

208.   Gauss's  Method  of  Substitution, 

In  solving  a  large  number  of  equations  simultaneously  it  is 
convenient  to  use  some  definite  system  of  eliminating  the  un- 
knowns, in  order  to  avoid  labor  and  the  danger  of  mistakes. 
Let  us  suppose  that  the  observation  equations  are  of  the  form 

d\x  +  biy  +  ciz  +  /i  =  vi, 


and  that  the  normal  equations  are  represented  by 
[aa]  x  +  [ab]  y  +  [ac]  z  +  [al]  =  o, 


o, 
o, 


>66] 


GAUSS'S  METHOD   OF  SUBSTITUTION  303 

. 
in  which  the  brackets  indicate  the  sum  01  all  the  terms  found 

by  multiplying  the  numerical  coefficients  according  to  the  rule 
on  p.  293. 

If  the  first  normal  equation  be  divided  by  [aa]  and  solved 
for  x,  the  result  is 


[aa]         [aa]         [aa] 
Substituting  this  in  the  second  equation,  we  have 


This  is  usually  abbreviated 

[bb  -  1]  y  +  [be  •  i]  z  +  [bl  •  i]  =  o.  [168] 

Substituting  this  in  the  third  equation,  we  have 

[be  •  i]  y  +  [cc  •  i]  z  -f  [cl  •  i]  =  o.  [169] 

These  two  equations,  [168]  and  [169],  are  called  the  "first 
reduced  normal  equations." 
Solving  [168]  for  y, 

[be  •  i]         [bl  •  i]  . 


[M-i]        [bb-i] 
whence  [cc  •  2]  z  +  [cl  •  2]  =  o,  [170] 

•  i     •      i  r  1  f  1  1  00    *    I  1    r-t  1 

in  which  [cc  •  2]  =  [cc  •  i]  —  7-  -  7  [fc  •  i] 

i] 


and 


The  solution  of  [170]  gives  the  value  of  z.  By  substituting 
this  in  [168]  and  [169]  the  value  of  y  may  be  found.  Finally, 
from  [166]  the  value  of  x  may  be  found. 

An  inspection  of  [166]  will  show  that  all  coefficients  below  and 
to  the  left  of  a  diagonal  drawn  from  the  x  term  of  the  first  equation 
to  the  z  term  of  the  third  equation  are  duplicates  of  the  others. 
These  may  be  omitted  in  writing  the  equations. 


304  ADJUSTMENT  OF  TRIANGULATION 

Solving  the  equations  in  Art.  207,  p.  302,  by  the  method  of 
substitution  just  described,  we  obtain  the  following  results: 

v2  =  +1.80, 

Z>3   =    -0.19, 

v5  =  -0.07, 

%   =    -0.75. 

Substituting  these  values  in  the  condition  equations,  p.  301, 
we  find  for  the  remaining  unknowns, 

*>i+8  =  +2.05, 
v7     =  +1.67, 

Z>4+5   =    -0.51, 
1>6        =    —0.83. 

The  final  angles  are  as  follows: 

1.  6i°o7'54."8o 

2.  38  28  36.  70 
3-   38  22  l8.  QI 

4.  42  01  ii.  71 

5.  2Q  14  32.  78 

6.  70  21  58.  37 

7.  49  26  23.  52 

8.  30  57  06.  35 

The  above  is  an  example  of  a  rather  unstable  solution  of  normal 
equations.  It  requires  a  relatively  large  number  of  significant 
figures  to  give  the  corrections  to  two  places  of  decimals. 

209.  Checks  on  the  Solution. 

In  practice  it  would  not  be  advisable  to  proceed  in  the  solution 
of  a  large  number  of  equations  without  some  safeguard  against 
mistakes  of  computation.  A  valuable  check  consists  in  adding 
to  the  normal  equations  an  extra  term  which  is  merely  the  sum 
of  all  the  coefficients  of  »i,  %,  etc.,  and  treating  this  term  like  any 
other  term  of  the  equation.  This  is  illustrated  later  in  the 
example  on  p.  313. 

210.  Method  of  Correlatives. 

When  there  are  many  condition  equations,  the  method  of  sub- 
stitution is  likely  to  prove  laborious.  If,  as  is  usually  the  case 
in  triangulation,  the  observations  are  direct  and  equal  in  number 
to  the  number  of  unknowns,  the  "Method  of  Correlatives  "  will 


METHOD   OF  CORRELATIVES  305 

be  found  preferable.  By  this  method  we  eliminate  one  unknown 
for  each  condition  equation,  employing  for  this  purpose  the 
method  of  undetermined  multipliers. 

Suppose  that  we  have  made  m  direct  observations,  MI,  M2, . . . , 
Mm,  of  m  different  quantities,  of  which  the  most  probable  values 
are 

2i    =   Mi   +  »i,      22    =   Af2   +%,..-,      2m   =   Mm  +  Vm. 

Let  these  m  unknowns  be  connected  by  the  following  n  con- 
ditions equations: 

a&i  +  a^Vz  .  .  .  amvm  +  /i  =  o, 

blVi   +  62%    •    •    .    &mflm  +  k    =   O, 


the  a's  being  the  coefficients  in  the  first  equation,  the  b's  those  of 
the  second,  etc.  The  quantities  /i,  k,  etc.,  represent  the  amounts 
by  which  the  observations  fail  to  satisfy  the  condition  equations. 
If  the  original  condition  equations  are  not  linear  in  form,  they 
must  be  made  so  by  a  method  similar  to  that  given  on  p.  299. 

Since  the  most  probable  values  of  the  v's  are  to  be  found,  we 
must  have 

fli2  +  *>22  +  •  •  •   =  a  minimum,  [172] 

or  PI  dvi  +  v2  dih  +  •  •  •   =  o  [172] 

for  all  possible  values  of  dvi,  d^  etc. 
Hence  it  must  hold  true  for  the  equations 

0i  dvi  +  02  dvz  +  •  •  •   =  o,  [173] 

&i  dvi  +  fe  dvz  +  •  •  •   =  o. 


obtained  by  differentiating  [171].  The  number  of  these  equa- 
tions is  n.  The  number  of  terms  in  [172]  is  m,  m  being  greater 
than  n.  Let  the  first  equation  in  [173]  be  multiplied  by  ki,  the 
second  by  k2,  etc.,  and  Equa.  [172]  by  —  i.  The  products  are 
then  added,  giving 

+  bih  +  •  •  •  -vi)  dvi 

•••   =o.     [174] 


306  ADJUSTMENT  OF  TRIANGULATION 

The  ^'s  are  to  be  so  determined  that  this  equation  will  hold  true. 
This  equation  will  be  satisfied  if  the  coefficient  of  each  differential 
in  it  is  placed  equal  to  zero,  that  is,  if 

-  •  •  •  «i  =  i 


Substituting  these  values  of  vi,  i>2,  etc.,  from  Equa.  [175]  in 
Equa.  [171],  we  obtain 


h  [aa]  +  k2  [ab]  +•••&«  [al]  +  /i  =  o, 
fe  W  +  k2  [bb]  +  •  •  -  kn  [bl]  +  h 


The  solution  of  these  equations  gives  the  values  of  ki,  fe,  k3, 
etc.,  which  are  the  correlatives  of  the  condition  equations.  By 
substituting  these  values  in  Equa.  [176]  the  v's  are  found.  Since 
the  form  of  Equa.  [176]  is  the  same  as  that  of  normal  equations, 
it  is  evident  that  they  may  be  solved  by  the  method  of  substitu- 
tion. 

In  case  the  observations  are  of  different  weight,  the  minimum 
equation  would  be 

Mi2  +  />2%2  +  •  •  •  pmVm2  =  a  minimum,  [177] 

and  the  other  equations  would  be  modified  accordingly. 

Example.  As  an  illustration  of  the  method  of  correlatives  we  will  use  the  same 
quadrilateral  that  was  adjusted  by  the  method  of  direct  elimination.  The  obser- 
vation equations  are  eight  in  number  and  all  of  the  form 


The  four  condition  equations  are 


(     fli+s  +  % 
tions  -j     vs  +  04+5 

I  V6  +  Z>6  + 


-  5.51     =  o, 
Angle  equations  \     v3  +  v*+b  +  v6  +  1.53     =  o, 

Z>6  +  V7  -\~  V 8  —  O.O2  =  O. 


Side  equation, 

2.65  V2  +  0.72  04-H,  +  3.51  V8  +  0.08  fi+8  —  2.66  vs  —  3-76  ^6  —  2.70  =  o. 


METHOD  OF  CORRELATIVES 

The  coefficients  of  the  corrections  are  then  tabulated  as  follows: 


307 


V. 

a. 

b. 

c. 

d. 

Vl+3 

+  1 

+0.08 

V2 

+  1 

+  2.65 

9f 

+  1 

+  1 

t>3 

+  1 

-2.66 

V<+5 

+  l 

+0.72 

fe 

+  1 

+  1 

fs 

+  1 

-3.76 

vs 

+  1 

+3-51 

Substituting  these  in  the  angle  equations, 


(i  +  8)  =  +2.058  -  0.094  X  0.08  =  +2.05 
2  =  +2.058  —  0.094  X  2.65  =  +1.81 
7  =  +2.058  —  0.406  =  +1.65 

+5.51    Check 


(4  +  5)  = 
6 


-0.435  +  2.66  X  —0.094  = 
-0.435  +  °-72  X  —0.094  = 
-0.435  —  0.406  = 


—0.19 
—0.50 
—0.84 
-1.53  Check 


5  =  —0.4064  —  3.76  X  —0.094  =  —0.05 

6  =  —0.435  ~~  0.406  =  —0.84 

7  =  +2.058  —  4.06  =  +1.65 

8  =      0.406  +  3.51    X  —0.094  =  —0.74 

+0.02    Check 

The  coefficients  of  the  correlative  Equa.  [176]  are  tabulated  as 
follows : 


aa. 

ab. 

ac. 

ad. 

bb. 

be. 

bd. 

cc. 

cd. 

dd. 

fl+S 

+  i 

+0.08 

0.0064 

V2 

+  i 

+  2.6S 

7.0225 

V7 

+  i 

+  1 

+1 

V3 

+  i 

-2.66 

7.0756 

V4+5 

+  i 

+0.72 

0.5184 

*>6 

+  i 

+1 

+1 

»| 

+1 

-3.76 

14.1376 

t>8 

+1 

+3-51 

12.3201 

Sum 

+3 

+  1 

+  2-73 

+3 

+1 

—  1.94 

+4 

-0.25 

+41.0806 

308  ADJUSTMENT  OF  TRIANGULATION 

The  correlative  equations  are  therefore 

I.  3  h  +  o  +  £3  +  2.73  ki  —  5.51  =  o 

II.   o  +  3  &  +  k3  —  1.94  £4  +  1-53  =  ° 

III.  ki  +  kz  +  4  ka  —  0.25  ki  —  0.02  =  o 

IV.  +2.73  ki  —  1.94  h  —  0.25  h  +  41-0806  h  —  2.7    =o 

The  solution  of  these  equations  gives  for  the  correlatives, 

ki  =  +2.058, 
kz  =  -0.435, 
£3  =  —0.406, 

£4  =    —0.094. 

Applying  these  equations  to  the  measured  angles,  we  obtain 
the  final  angles. 

Measured  Angles.        Correction.        Seconds  Corrected. 

(i  +  8)  92°  04'  59"-io  +  2".o5 
2  38  28  34  .90  +  i  .81 
7  49  26  21  .85  +  i  .65 


oo  .00  Check 


3    38°  22'  i9".ia  -  o".i9         i8".9i 

(4  +  5)  71  15  45  .00-0  .50         44  .50 

6   .  70  21  59  .20  —  o  .84         58  .36 


oi".77 
e"  =  oi  .77 

oo  .00  Check 


5  29°  14'  32".8s  -  o".oS  32".8o 

6  70  21  59  .20  —  o  .84  58  .36 

7  49  26  21  .85  +  i  .65  23  .50 

8  3°  57  07  .10  —  o  .74  06  .36 


or  .02 
e"  =  oi  .02 

oo  .00  Check 


The  check  of  the  side  equations  is  as  follows: 

2.65  X      1.81  =      4.80  —0.08  X      2.05  =  — 0.16 

0.72  X  —0.50  =  —0.36  +2.66  X  —0.19  =  —0.49 

3.51  X  —0.74  =  —2.60  +3-76  X  —0.05  =  —Q.20 


+1.84  -0.85 

-0.85 

+2.69 

(Should  equal  2.70.) 


METHOD   OF  DIRECTIONS 


309 


If  the  sums  of  the  log  sines  are  again  computed  (see  p.  301), 
using  the  corrected  seconds,  they  will  be  found  to  equal  9.481  5090 
for  both  columns. 

211.   Method  of  Directions. 

The  method  of  correcting  the  directions  instead  of  the  angles  is 
particularly  applicable  when  the  measurements  have  been  taken 
by  the  method  of  directions,  Art.  43.  In  the  United  States 
Coast  Survey  office  it  is  the  usual  practice  to  employ  this  method 
of  adjusting,  whether  the  observa- 
tions were  made  by  the  direction 
method  or  by  the  method  of  repe- 
tition. 

In  the  quadrilateral  adjusted  in 
Arts.  208-210,  let  us  denote  the 
directions  by  the  numbers  i  to  12 
(Fig.  114)  and  the  corrections  to 
those  directions  by  the  same  num- 
bers,    (i),    (2),    etc.,    enclosed   in 
parentheses.     Each    angle    is    ex- 
pressed as  the  difference  of  two  directions;  that  is,  the  angle 
—  4  +  5  means  the  angle  between  the  directions  marked  4  and  5. 
The  four  condition  equations  are  the  same  as  before  except  as 
to  the  change  in  notation. 


Angle  Equations 


10 


FIG.  114. 


1-53  =  o. 
-   (4)+  (6) -0.02=0. 


Side  equation, 

-5.31  (n)  +  2.65  (12)  +  3.04  (7)  +  0.72  (9)  -  3.51  (2) 
+  3-59  (3)  -  0-08  (i)  +  2.66  (10)  -  3.76  (8)  -  2.7  =  o. 

If  CD  were  a  fixed  line  obtained  by  a  previous  adjustment,  the 
corrections  (9)  and  (10)  would  be  omitted.  The  angle  equations 
could  be  simplified  in  this  case  by  selecting  two  equations  which 
involve  angles  depending  upon  those  two  directions. 


3io 


ADJUSTMENT  OF  TRIANGULATION 


The  first  table  for  the  coefficients  of  the  corrections  is  given 
below. 


Direction. 

a. 

b. 

C. 

d. 

I 

—  I 

+0.08 

2 

—  .  J 

-3-51 

3 

+  1 

+1 

+3-59 

4 

—  I 

. 

—  I 

+  1 

—  I 

+1 

7 

—  I 

""""  I 

-r3-°4 

8 

+1 

-3.76 

9 

+  1 

+0.72 

10 

—  I 

+  2.66 

ii 

—  I 

+  1 

-5-31 

12 

+I 

+  2.65 

The  remainder  of  the  work,  that  is,  the  calculation  of  co- 
efficients ^aa,  ^ab,  etc.,  and  the  solution  of  the  numerical  equa- 
tions, is  carried  out  as  in  the  preceding  example  (Art.  210).  The 
solution  of  the  normal  equations  gives  the  corrections  to  the 
directions.  The  correction  to  any  angle  is  the  difference  of  the 
corrections  to  the  directions  of  its  sides. 


212.  Adjusting  New  Triangulation  to  Points  already  Adjusted. 

In  the  quadrilateral  shown  in  Fig.  115  the  triangle  BDE  is  sup- 
posed to  have  been  previously  adjusted.  Point  C  is  determined 
by  the  directions  i,  2,  and  3  in  connection  with  the  directions 
along  the  sides  of  the  fixed  triangle,  and  also  by  directions  4,  5, 


ADJUSTING  NEW  TRIANGULATION 


311 


and  6.     The  directions  to  be  found  are  i,  2,  3,  4,  5,  and  6.     The 
directions  as  taken  from  the  field-notes  are  as  follows: 


Point  sighted. 

Direction  after  local 
adjustment. 

Corrected  seconds. 

AtC   ' 

0                /                  // 

D 

0   00   00.00 

B 

123  49  24.97 

E 

207  52  33-50 

>  "    B     At  D 

0               /                    //• 

„ 

A 

o  oo  oo.oo 

00.67 

C 

296  57  55-83 

E 

311    12    14.48 

12  .69 

B 

258    27    57.39 

57-i8 

AtE 

. 

0                /                    // 

// 

D 

o  oo  oo.oo 

01.32 

C 

13  38  27.54 

B 

81  28  43-98 

43-05 

AtB 

F 

o  oo  oo.oo 

01.  06 

E 

122    32    11.29 

12.56 

C 

150   38   41.62 

D 

168  19  14.81 

15-48 

In  taking  directions  from  this  table,  the  corrected  seconds 
should  be  used  whenever  an  adjustment  has  been  made. 

The  number  of  angle  equations  in  the  figure  is  _/J— ~s  +  i,  or 
6  —  4  +  1  =3.  The  number  of  side  equations  is  /  —  2  s  +  3, 
or  6  —  8+3  =  1.  Since,  however,  the  exterior  triangle  is 
already  adjusted,  there  will  be  but  two  angle  equations  needed 
in  the  adjustment.  For  these  two  angle  equations  take  the 
triangles  DCE  and  BEC\ 

then  -(a)  +  (i)  -  (3)  +  (/)  -  (6)  +  (4)  =  o 

and  -  (i)  +  (b)  -  (5)  +  (6)  -  (c)  +  (2)  =  o. 


312  ADJUSTMENT  OF  TRIANGULATION 

But  since  the  exterior  lines  are  not  to  be  changed,  (a),  (/), 
(b),  and  (c)  are  all  zero. 

The  absolute  terms  in  the  angle  equations  are  found  as  fol- 
lows: 

-(a)  +  (i)  i3038'26".22 
-(3)  +  (/)  14  14  16  .86 
-(6)  +  (4)  152  07  26  .50 

180  oo  09  .58 
180  oo  oo  .02 


-d)  +  (&)  67°59'i5".5i 
-(5)  +  (6)  84  03  08  .53 
-(c)  +  (a)  28  06  29  .06 

J79  59  S3  -io 

180  oo  oo  .08 

+6".98 


For  the  side  equation  take  the  pole  at  C. 


sin  (-(2)  +  (<*))    sin(-(i)  +  (6))    sin  (  -.(3)  +  (/)) 
sin(-(*)  +  (3))  'sin  (-(c)  +  (2))  'sin(-(a)  +  (i))  " 

Tabulating  the  log  sines, 

log  sin  (+)  dit.  I" 

-(2)  +  (rf)  I7°4o'33".86  9-4823521  +66.I 

-(i)  +  (6)  67  50  15  .51  9.9666666  +  8.6 

-(3)  +  (0  U  14  16  .86  9.3908478  +83.0 

8.8398665 

log  sin  (—  ) 

-(«)  +  (3)  38°  29'  s8".6S    9.7941460  +26.5 

—  (c)  +  (2)  28  06  29  .06    9.6731464  +39.5 

—  (a)  +  (i)  13  38  26  .22    9.3726010  +86.7 

8.8398934 
8665     . 

constant  =        —269 
The  side  equation  is  therefore 

+6.61  X  -  (2)  +  0.86  X  -  (i)  +  8.30  X  -  (3) 

-  2.65  X  (3)  -  3.95  X  (2)  -  8.67  X  (i)  -  26.9  =  o. 

Carrying  out  the  same  process  as  outlined  in  Art.  210,  we  have 
the  following: 


ADJUSTING  NEW  TRIANGULATION 
TABLE  OF  COEFFICIENTS. 


313 


Direc- 

a. 

b. 

c. 

Sum. 

aa. 

ab. 

ac. 

as. 

bb. 

be. 

bs. 

cc. 

cs. 

tion. 

I 

+i 

—i 

-  9-53 

-  9  53 

+i 

—i 

-  9-53 

-  9-53 

+i 

+  9-53 

+9-53 

90.8209 

90.8209 

2 

+i 

—  10.56 

-  9.56 

+i 

—10.56 

-9-56 

111.5136 

100.9536 

3 

—i 

-10.95 

-II-9S 

+i 

+10.95 

+11.95 

119.9025 

130.8525 

4 

+i 

+  i 

+i 

+  i 

5 

—i 

—  i 

+i 

+i 

6 

—i 

+i 

0 

+i 

-i 

+i 

Total.... 

+4 

—  2 

+1.42 

+3-42 

+4 

-1.03 

+0.97 

322.2370 

322.6270 

From  these  sums  we  derive  the  correlative  equations. 
CORRELATIVE  EQUATIONS 


Number. 

*> 

* 

*, 

Const. 

Check. 

Sum. 

I 
2 

3 

+4 

—  2 

+4 

+     1-42 
-     1-03 
+322.24 

+  9-56 
-  6.98 
—  26.9 

+   12.98 
—     6.01 
+295.73 

+     3-42 
+     0.97 
+322.63 

It  should  be  observed  that  the  "constant"  terms  are  taken 
directly  from  the  condition  equations.  The  "sum"  term  con- 
tains the  sum  of  the  coefficients  of  the  &'s.  The  "check" 
term  is  the  algebraic  sum  of  the  constant  and  sum  terms.  The 
solution  is  given  in  detail  in  the  following  table:  The  different 
operations  are  indicated  in  the  left-hand  column.  The  factors 
by  which  the  equations  are  multiplied  are  in  the  right-hand 
column. 


2 

+4 

-1.03 

-6.98 

—  6.01 

Factor 

IX   -  — 

—  i 

+0.71 

+4-78 

+6.49 

+1 

4 

2 

II 

+3 

-0.32 

—  2.20 

+0.48 

3 

+322.24 

—  26.9 

+  295-73 

IX      '  42 

—     0.50 

-  3-39 

-       4.6l 

-0-355 

4 

3 

—     0.03 

-  0.24 

+       0.05 

III 

+321.71 

-30.53 

+  291.17 

ADJUSTMENT  OF  TRIANGULATION 


The  preceding  table  is  an  abbreviated  form  of  the  method  of 
substitution  explained  in  Art.  208. 
The  correlatives  are  found  as  follows: 


I. 

II.- 

III. 

Const. 
k, 
kz 
*i 

+9-56 
+0.135 
-1.487 

—  2.20 
—  0.03 

-30.53 

,      _    30.53    _     , 

'         32L70 

£2  =  123  =  +0.7433 
t        8.208 

2  .230 

R\                       —        2.OS2 

+8.208 

Calculating  the  corrections  for  the  correlatives, 


I. 

2. 

3- 

4« 

5- 

6. 

fel 

—  2.052 

+  2.052 

—  2  .052 

+  2.052 

k, 

-  o  .  743 

+  0-743 

-0-743 

+0-743 

k3 

-0.904 

—  I  .OO2 

-1.039 

-3.699 

—  O.262 

+  I.OI3 

-2.052 

-0-743 

+  2-795 

Applying  these  corrections  to  the  directions,  we  have  the 
final  adjusted  values 


Dir.  No. 

Observed  directions. 

Correction. 

Corrected  seconds. 

4 

01                        II 

O   OO   OO.OO 

// 

-2.05 

// 

57-95 

6 

123  49  24.97 
207  52  33-50 

-0-74 
+  2.80 

24.23 
36.30 

I 

2 

3 

13  38  27.54 
150  38  41.62 
296  57  55-83 

—  3.70 
—  0.26 
+  I.OI 

23.84 
41.36 
56.84 

213.  The  Precision  Measures. 

Referring  to  the  equation  of  the  curve  of  error,  Art.  197, 

y  =  ke-h**9  [149] 


THE  PRECISION  MEASURES  315 

we  see  that  there  are  two  constants  to  be  determined  for  any 
particular  set  of  observations.  These  two  constants  are  not 
independent,  however,  as  will  be  shown.  The  total  area  be- 
tween the  curve  and  the  X  axis  was  taken  equal  to  unity;  there- 
fore 


ft 

or  k  I     e~ 

JQ 


~h*xZ 


Q 
from  which  f°  e 

JQ  2k 

In  order  to  integrate  this  expression  let  /  =  hx  and  dt  =  h  dx. 

Then  f  e~*dt  =   f   e~h^hdx. 

JQ  JQ 

Multiplying  this  equation  by    . 

f~  e~*dt=   f  e~htdh, 
JQ  JQ 

we  have 

2  rf"      /V*> 

=    /      /     e-h*(l+x*>hdxdh 
Jo    JQ 

dx  C  e~"  (1+xl)  (  -  2  A)  (i  +x2)  dh 


=  -  r  — —*  =  -[tan-1*]   =  - 

2  J0      I  +  X2         2  L  Jo         4 

Therefore  f  e~*dt  =  — 

JQ  2 

V*      h 
and  =  -"7' 

2  2Jfe 


or  K  =  — =>  [178] 

which  shows  the  relation  between  the  two  constants. 


316  ADJUSTMENT  OF  TRIANGULATION 

The  equation  of  the  curve  of  error  may  now  be  written 

y  =  ^7=e-»*\  [179] 

VTT 

214.  The  Average  Error. 

The  average  error  (r/)  is  the  arithmetical  mean  of  the  errors, 
all  taken  with  the  same  sign.  To  derive  an  expression  for  the 
average  error,  we  see  from  equation  (142)  that  /  (x)  dx  is  the 
probability  that  an  observation  will  fall  between  the  limits  x 
and  x  +  dx\  that  is,  it  represents  the  proportion  of  all  the  errors 
that  will  probably  fall  within  these  limits.  Hence,  if  n  observa- 
tions are  made,  the  number  in  this  strip  will  be  nf  (x)  dx.  The 
sum  of  all  the  observations  will  be 


or  2n 

ro 


n  I     xf(x)dx, 

«/_00 

,00 

/     xf(x)dx. 
Jo 


The  average  error  equals  the  sum  of  the  errors  divided  by  the 
number,  that  is, 


C"  vf( 

J0    XJ(X 

_2_y 

^7 


dx 


(-2h2x)dx 


215.  The  Mean  Square  Error. 

The  mean  square  error  (/*)  of  an  observation  is  the  square 
root  of  the  arithmetical  mean  of  the  squares  of  the  errors.  Since 
the  number  of  errors  between  x  and  x  +  dx  is  nf  (x)  dx,  the 
sum  of  the  squares  of  these  errors  is 

nx2f(x)dx.     • 


THE  PROBABLE  ERROR  317 

The  sum  of  the  squares  of  all  the  errors  is 
n  F  *?f(x)dx. 

*J   —  00 

Therefore  n?  =  -^  f    e~hSxt^dx.  (d) 

VTT  ^-°o 

-^  /      e~h2x2dx  =  i,    or 


If  we  differentiate  this  with  respect  to  h  as  the  independent 
variable,  we  obtain 

-2k  ("  f**a*dx=-~-  (e) 


Substituting  (e)  in  (d), 

M  =  -V  [181] 

h  V2 

216.  The  Probable  Error. 

The  probable  error  (r)  of  an  observation  is  an  error  such  that 
one  half  the  errors  of  the  series  are  greater  than  it  and  the  other 
half  are  less  than  it;  that  is,  the  probability  of  making  an 
error  greater  than  r  is  just  equal  to  the  probability  of  making 
an  error  less  than  r. 

The  probability  that  an  error  of  an  observation  will  fall  be- 
tween the  limits  x  and  x  +  dx  is  /  (x)  dx.  The  probability 
that  the  error  will  fall  between  the  limits  +r  and  —  r  is  given 
by 

r2  h    /'+r 
/(*)</*  =  -=/   e~ 
V  7T  t/0 

by  the  definition. 
To  integrate,  let  /  =  hx,  and  dt  =  h  dx, 

Then  ^=1     e~r'dt  =  -. 


If  we  evaluate  this  integral  for  assumed  values  of  hr  and 


318  ADJUSTMENT  OF  TRIANGULATION 

then  interpolate  for  the  value  of  hr  corresponding  to  J,  we 
find  it  to  be  0.47694. 


Therefore  r  =  24Z22*.  [l82] 

All  the  precision  measures  have  now  been  expressed  in  terms 
of  /?,  and  it  is  evident  that, 

r  =  0.8453 V  [183] 

=  0.6745  M.  [184] 

The  mean  square  error  (#)  is  the  largest,  and  the  probable 
error  (r)  is  the  smallest,  of  the  three  precision  measures. 

Any  one  of  the  three  precision  measures  may  be  used  to  com- 
pare the  relative  accuracy  of  different  series  of  observations, 
provided  the  different  series  are  made  under  the  same  condi- 
tions, so  as  to  be  affected  by  the  same  constant  errors.  In 
Europe  the  mean  square  error  has  been  used  more  than  the 
probable  error;  in  the  United  States  the  probable  error  is  gen- 
erally employed.  There  are  some  advantages,  however,  in  the 
use  of  the  average  error  (17).  Theoretically  it  is  slightly  less 
accurate  than  either  of  the  others;  but  inasmuch  as  the  quan- 
tity itself  is  an  estimate  of  an  uncertainty  in  measurement, 
this  objection  is  not  a  serious  one.  The  value  of  r\  lies  between 
the  values  of  ju  and  r.  The  method  of  computing  77  is  simpler, 
as  will  be  shown  later,  than  the  computation  of  either  /*  or  r. 

Since  in  Equa.  [158]  it  was  shown  that  p  varies  as  h2,  it  follows 
that 


that  is,  the  weights  of  the  different  observations  on  a  quantity 
vary  inversely  as  the  squares  of  the  precision  measures. 

If  M  is  the  precision  measure  of  a  direct  observation  of  weight 
i,  and  MO  is  the  precision  measure  of  the  mean,  then  since  the 
weight  of  the  mean  is  n,  the  number  of  observations, 

«=-*=•  co 

v« 


COMPUTATION  OF  THE  PRECISION  MEASURES  319 

217.   Computation  of  the  Precision  Measures. 

Direct  Observations  of  Equal  Weight.  To  find  M,  the  mean 
square  error  of  an  observation,  suppose  that  we  have  n  direct 
observations  of  equal  weight  made  on  a  quantity  M ,  and  that 
the  results  are  M\t  MI,  .  .  .,  and  that  MQ  is  the  most  probable 
value.  Let  the  errors  be  Xi,Xz,  .  .  .  and  the  residuals  Vi,vz,  .  .  .  . 

Then  in  this  case  the  residuals  are 

Vi  =  Mi  -  Mo, 

1)2   =  M2  —  MQ 


and 


If  MQ  were  the  true  value  of  M ,  the  residuals  would  be  the 
same  as  the  true  errors,  and  in  that  case 


[186] 
n 

But  in  any  limited  number  of  observations  this  is  not  suffi- 
ciently exact.     To  obtain  a  more  accurate  expression,  place 

MQ  +  XQ  =  M; 
then 

Xi  =  Mi  -  (Mo  +  XQ)   =  Vi  -  XQ, 
£2  =  M 2  -  (M0  -f  XQ)   =  Vz  -  XQ, 


Squaring,  adding,  and  dividing  by  n, 


n 
Since  ^v  =  o,  Art.  195,  this  reduces  to 


n  n 

The  real  value  of  x0  is  unknown;  it  may  be  taken  as  approxi 


320  ADJUSTMENT  OF  TRIANGULATION 

mately  equal  to  the  mean  square  error  of  M0,  which,  from 
Equa.  (/),  is 

*>=-^;  [187] 

Vn 
whence 


Therefore  /i  =  y 


n  —  i 


To  find  no,  the  mean  square  error  of  the  mean  value,  we  have, 
by  Equa.  (/), 


mffZ$  Il89l 

From  Equa.  [184], 


y 
/•  =  0.6745  V  -^ —  [190] 


ro  =  0-6745' 


and 

To  find  the  average  error  (t;)  of  a  single  observation,  we  see  that, 
from  Equa.  [188], 


n 
On  the  average  the  values  of  these  residuals  will  be 


Adding  and  dividing  by  n, 

£,v      ./n  —  i    ^x      .In  —  \ 

-  =  y • fs—  =  v ~  •  >?. 

n  n         n  n 


OBSERVATIONS  OF  UNEQUAL  WEIGHTS  321 


Therefore  rj  =        ~*         ,  [192] 

v  n  (n  -  i) 


and  i?o  = =^='  ^93] 


The  probable  error  is  sometimes  computed  from  the  average 
error  in  order  to  avoid  computing  the  squares  of  the  residuals. 
From  Equa.  [183], 

0-8453  Xy 

r  =     ,    .     *    ,  [i94] 

v  n  (n  —  i) 


and  *-.  [195] 

w  Vw  —  i 

Evidently  the  mean  error  may  also  be  computed  from  77. 
218.   Observations  of  Unequal  Weights. 
If  the  observations  have  unequal  weights,  let  pi,  p%,  etc.,  be 
the  weights;  then 

»  MI  =  -=  ,    etc. 


By  Art.  199,  if  each  observation  is  multiplied  by  the  square 
root  of  its  weight,  the  observations  are  all  reduced  to  weight 
unity.  The  residuals  are  therefore 

etc. 


Applying  Formulae  [188]  to  [195]  to  these  residuals,  we  have 


322 


ADJUSTMENT  OF  TRIANGULATION 


Also, 


from  which 


r  =  0.6745 


n  —  i 


=  0.6745 


770 


[200] 
[201] 

[202] 
[203] 
[204] 


r  =  0.8453  I,  [205] 

ri  =  0.8453  n,  [206] 

r0  =  0.8453  w-  [207] 

219.  Precision  of  Functions  of  the  Observed  Quantities 
Suppose  that  a  quantity  M  is  denned  by 

M  =  M i  +  Afa, 

where  Ifi  and  M2  are  independent  and  are  observed  directly. 
Let  the  mean  square  error  (m.s.e.)  of  M\  be  MI>  and  let  that  of 
M2  be  M2,  the  m.s.e.  of  the  function  M  being  denoted  by  pp.  If 
we  suppose  the  errors  in  the  determination  of  MI  to  be  Xi,  Xi", 
Xi",  .  .  .  ,  and  those  of  M%  to  be  x^ ,  xj' ,  fy'",  .  .  .  ,  then  the 
real  errors  of  If,  computed  from  the  separate  observations  on  MI 
and  Af2,  will  be 


PRECISION  OF  FUNCTIONS  OF  THE  OBSERVED  QUANTITIES       323 

and        ;-y_fa'±  *•)'  +  (*"  +*"?+•••       '       '' 

n 

Xi2  +  2         XiXz  +         #22 


But  the  XiXz  terms  will  cancel  out,  because  in  the  long  run 
there  will  be  as  many  -f-  as  —  products  x\x^  of  the  same  magni- 
tude. 

Therefore  MF2  =  Ml2  +  rf.  [208] 

From  Equas.  [183]  and  [184]  it  is  evident  that 

rF2  =  r?  +  rf  [209] 

and  7jF2  =  77i2  +  rfe2.  [210] 

Let  us  suppose  that  the  function  is  denned  by 

M  =  aiMi, 
where  di  is  a  constant;  then  the  real  errors  of  M  will  be 


>    •        •    > 

0125X2 
and  ui/  =  — ^* — -  =  /iv  ui  . 


or  MF  =  fllMl.  [211] 

By  combining  [208]  with  [211]  it  is  clear  that  if 
M  =  aiMi  -f  azMz  +  dzMz  +  •••-, 

then  rf  =  5aV,  [212] 

[213] 

[214] 
Suppose  that  the  function  is  of  the  general  form  indicated  by 


Let  M  i  =  0i  -f  Wi,  Af2  =  (h  +  ^2,  etc.,  in  which  ai  is  a  close 
approximation  to  Afi,  02  is  a  close  approximation  to  Jl/2,  and  Wi 
and  mz  are  small  corrections  such  that  their  squares  may  be 
neglected.  We  may  regard  mi  and  ^2,  etc.,  as  containing  the 
real  errors  of  Afi,  Af2,  .  .  .  ,  and  A*I,  M2,  •  •  •  may  be  considered 


324  ADJUSTMENT  OF  TRIANGULATION 

as  the  mean  square  errors  of  mi,  nh,  etc.     Substituting  in  (g), 
we  have 

M  =  /  (Oi  +  mi),  (<h  +  nh)  .  .  .  ). 

Expanding  this  function  by  Taylor's  theorem  and  denoting 


it       ™>  .  .  n\ 

M  =  M'+nii-  --  \-nh-  --  h  •  •  •  ,  (ft) 

0#i  OO2 

in  which  the  terms  containing  the  squares  and  higher  powers  of 
mi,  nh,  .  .  .  have  been  omitted.     Then  the  m.s.e.  of  M  is  the 
same  as  the  m.s.e.  of  the  terms  in  (/?). 
By  Equa.  [212],  this  is 


or,  with  sufficient  accuracy, 

JdMj  ,      2[dMj  , 

^=4wJ+M2LaM2J  +  -'--  [2I5] 

Similarly, 

, 

+  -"'        [2I6] 


2  ,  , 

and  [2I7] 


.  It  should  be  observed  that  in  the  preceding  cases  the  unknowns 
are  supposed  to  be  independent  of  each  other.  If  the  quantities 
MI,  M%,  etc.,  are  functions  of  the  same  variable,  a  different  pro- 
cedure is  necessary. 

Also,  in  case  the  unknowns  are  subject  to  any  number  of  con- 
ditions, the  computation  of  the  precision  measure  of  any  function 
must  be  so  modified  as  to  take  into  account  the  effect  of  these 
conditions. 

220.  Indirect  Observations. 

The  computation  of  the  precision  of  the  adjusted  values  in  the 
case  of  indirect  observations  is  more  complicated  than  in  the 


CAUTION  IN  THE   APPLICATION  OF  LEAST  SQUARES     325 

case  of  direct  observations,  because  it  is  necessary'  to  know  the 
weight  of  each  of  the  unknowns,  and  this  can  only  be  found  by 
the  solution  of  equations  similar  to  the  normal  equations. 

It  may  be  shown  that  if  there  are  n  observations  on  q  un- 
knowns, ther 

=  V  -2£_,  [2l8] 

n-q 

where  /*  is  the  m.s.e.  of  an  observation  of  weight  unity. 

If  pz  is  the  weight  of  an  unknown,  then  the  m.s.e.  of  this  un- 
known is 

/ X~*    o 

[219] 

Similarly,  r  =  0.6745  V  ^"    >  [220] 


and  77  =        **  [222] 

[223] 

hn  (ti  -  q) 

221.   Caution  in  the  Application  of  Least  Squares. 

In  applying  the  preceding  principles  it  should  be  kept  in  mind 
that  the  ordinary  adjustment  by  the  method  of  least  squares 
deals  with  the  accidental  errors  only  and  can  tell  us  nothing 
about  the  constant  or  systematic  errors  which  may  affect  the 
results  of  observation.  The  "  probable  error  "  may  therefore  be 
far  from  the  true  error  because  such  constant  errors  are  present. 
We  should  think  of  the  precision  measures  as  indicating  the  de- 
viation of  the  result  from  the  mean  result  of  a  large  number  of 
such  observations,  rather  than  its  deviation  from  the  true  value. 
It  is  usually  true  that  the  constant  or  the  systematic  errors  are 


326  ADJUSTMENT  OP  TRIANGULATION 

far  more  serious  than  the  accidental  errors;  the  observer  should 
be  continually  on  the  watch  for  constant  errors  which  may  affect 
his  result.  So  long  as  the  conditions  under  which  a  measure- 
ment is  made  remain  exactly  the  same  the  systematic  errors  are 
likely  to  be  the  same  and  are  therefore  not  observed.  The 
presence  of  such  errors  is  most  likely  to  be  observed  when  the 
conditions  are  varied  as  much  as  possible.  If  observations  are 
made  at  different  temperatures,  or  under  different  conditions  of 
illumination,  or  with  different  instruments,  the  variations  of  the 
results  are  usually  greater  than  when  the  conditions  are  not 
changed.  These  variations  indicate  the  presence  of  systematic 
errors  and  often  enable  the  observer  to  estimate  their  magnitude. 

The  computation  of  the  most  probable  value  improves  the 
result  with  respect  to  the  accidental  errors,  but  leaves  the  more 
serious  form  of  error  untouched.  The  futility  of  multiplying 
observations  and  adjusting  them  for  the  purpose  of  removing  the 
small  accidental  errors,  and  at  the  same  time  failing  to  remove 
the  large  constant  error,  may  be  illustrated  by  the  results  ob- 
tained by  a  marksman  who  holds  his  rifle  steadily  and  places  all 
his  shots  in  a  small  group,  but  whose  rifle  sights  are  so  far  out  of 
alignment  that  his  shots  all  strike  far  from  the  bull's-eye.  Of 
what  use  is  the  large  number  of  shots  under  those  circumstances? 
An  adjustment  of  his  results  by  least  squares  would  correspond 
to  an  attempt  to  find  the  center  of  his  group  of  shots,  and  would 
tell  nothing  about  the  distance  from  the  bull's-eye.  A  study  of 
the  causes  of  the  error  so  that  he  could  make  an  adjustment  of 
his  sights  would  accomplish  more  toward  hitting  the  mark  than 
an  infinite  number  of  shots  find  under  the  original  conditions. 
Of  course  the  comparison  is  quite  untrue  in  one  respect;  the 
marksman  knows  where  his  mark  is,  while  the  observer  can  never 
know  the  true  value  of  the  quantity  he  is  measuring. 

While  the  method  of  least  squares  may  not  show  directly  the 
presence  of  constant  errors,  a  study  of  the  precision  of  the  results, 
and  a  knowledge  of  the  law  governing  the  behavior  of  accidental 
errors,  may  enable  the  observer  to  detect  the  presence  of  constant 


CAUTION  IN  THE  APPLICATION  OF  LEAST  SQUARES     327 

error,  or  at  least  to  decide  whether  it  is  probably  present,  and 
consequently  to  so  modify  his  methods  of  observing  as  to  reduce 
the  effect  of  such  constant  error.  Variations  in  the  result  which 
are  greater  than  the  error  of  observation  shown  by  the  precision 
measures  is  likely  to  mean  that  systematic  error  is  present.  This 
tracing  of  errors  to  their  sources,  and  the  consequent  modification 
of  instruments  and  methods,  may  constitute  the  most  important 
application  of  least  squares. 

REFERENCES 

Following  are  a  few  references  to  extended  works  on  the  sub- 
ject of  Least  Squares. 

BARTLETT,  The  Method  of  Least  Squares  (an  Introductory  Treatise). 

CHAUVENET,  Treatise  on  the  Method  of  Least  Squares.  (Theory  —  Applications 
to  Astronomy.) 

CRANDALL,  Geodesy  and  Least  Squares.     (Applications  to  Geodesy.) 

MERRIMAN,  Treatise  on  the  Method  of  Least  Squares. 

UNITED  STATES  COAST  AND  GEODETIC  SURVEY,  Special  Publication  No.  28.  (Prac- 
tice of  the  United  States  Coast  and  Geodetic  Survey.) 

WRIGHT  AND  HAYFORD,  Adjustment  of  Observations.     (Applications  to  Geodesy.) 


PROBLEMS 

Problem  i.    The  following  angles  are  measured  at  station  O. 

AOB  =  31°  10'  i5".6  weight  (i) 

BOC  =  19  21  17  .4  "  (i) 

AOC  =  50  31  33  .5  "  (2) 

COD  =  38  50  16  .o  "  (2) 

BOD  =  58  ii  32  .o  "  (i) 

AOD  =  89  21  51  .5  «  (i) 
Adjust  the  angles. 

Problem  2.    The  angles  of  a  triangle  are  as  follows: 

A  53°53'38".94  wt.  (3) 
B  79  22  56  .17  «  (4) 
C  46  43  29  .27  "  (2) 

The  spherical  excess  is  2". 83. 
Adjust  the  triangle. 


328  ADJUSTMENT  OF  TRIANGULATION 

Problem  3.  The  angles  of  a  quadrilateral  are  as  follows,  the  numbers  correspond- 
ing to  those  in  Fig.  113.  The  weights  are  all  unity.  The  spherical  excess  may  be 
neglected. 

1.  23°3l'l2".5 

2.  37  01  22  .5 

3-  67  35  38  -3 

4.  51  51  26  .7 

5.  29  56  50  .o 

6.  30  35  33  -2 
7-  72  37  35  -o 
8.  46  49  47  -5 

The  sum  angles  are 

8  +  1  7o°2i'os".o 

2  +  3  104  37  oo.  o 

4  +  5  81  48  20  .8 

6  +  7  103    13  08  .4 
Adjust  the  quadrilateral. 


FORMULA  AND   TABLES 


FORMULA 

SERIES 


a*       *•       *« 
_+___ 


=  *  +  ?  +  —  +  —  +• 
6        40        112 

~3         ,r5         r7 

tan-*  *  =  *-  -  +  ---• 
35        7 

BINOMIAL  THEOREM 

(a  +  6)«  =  aw  +  wa"1-^  +  W  (^2~  ^  gm 

MACLAUREN'S  THEOREM 


TAYLOR'S  THEOREM 

#  & 

V      V 

LOGARITHMIC   SERIES 


OTHER  SERIES 

i 


i  —  x 

i 


330 


FORMULA  331 

ELLIPSE   AND   SPHEROID 
*-*^. 


Ra 


-«*sins40* 
NRm 


N  cos2  a  +  Rm  sin2  a 
Mean  radius  =  p  =  ^NRm- 

CONSTANTS 

logio  X  =  M  \Oge  X. 

M  =  modulus  of  system  of  common  logarithms 

=  0.434  2945. 
log  M  *=  9.637  7843- 

•K  =  3.141   592  65.     log  =  0.497  1499. 

*—  =  57.29577.         log  =  1.758  1226. 

180°  X  60' 

=  3437.747.          log  =  3.536  2739 

180°  X  60'  X  60 


206  264.8.         log  =  5.314  4251. 

(Approx.) 


IT 

III 


arc  i"       sm  i         tan  i 
arc  i"  =  o.ooo  004  848  137.         log  =  4.685  5749. 

— 77  =  206  264.806  =  number  of  seconds  in  the  radian, 
arc  i"  =  about  0.3  inch  at  distance  of  one  mile. 


CLARKE  SPHEROID   (1866) 

a  =  6  378  206.4  meters.  log  =  6.804  6985. 

6  =  6  356  583-8  meters.  log  =  6.803  2238. 

(Clarke's  value  of  meter,  3.280  8693  feet.) 

a  =  6  378  276.5  legal  meters.  log  =  6.804  7033. 

b  =  6  356  653.7  legal  meters.  log  =  6.803  2285. 

(Q.  S.  legal  meter,  39.37  inches  or  3.280  8333  feet.) 


332  FORMULAE 

COAST  SURVEY  SPHEROID    (1909) 

0  =  .6  378  388  ±  18  meters. 
j  =  297.0  ±0.5. 
b  =  6  356  909  meters. 

RELATION  BETWEEN  tJNITS  OF  LENGTH 

(Legal)  Meters  in  one  foot  =  0.304  8006.  log  =  9.484  0158. 

Feet  in  one  (legal)  meter     =  3.280  8333.  log  =  0.515  9842. 

Inches  in  one  (legal)  meter  =  38.37. 


TABLES 


333 


TABLE  I.  — TABLE  FOR  DETERMINING  RELATIVE  STRENGTH 
OF  FIGURES   IN   TRIANGULATION 


!  o 

10° 

12' 

14° 

16° 

18° 

20° 

2'2' 

24  : 

»• 

28C 

30° 

35° 

40  -' 

4.5° 

50' 

U* 

BO* 

85* 

70  c 

7.5  c 

»• 

s:/j 

90° 

10 

428 

359 

12 

359 

295 

253 

14 

315 

253 

214 

187 

16 

284 

225 

187 

162 

143 

18 

262 

204 

168 

143 

126 

113 

20 

245 

189 

153 

130 

113 

100 

\il 

22 

232 

177 

142 

11 

103 

91 

81 

74 

24 

221 

167 

134 

111 

95 

83 

74 

67 

61 

26 

213 

160 

126 

104 

89 

77 

6s 

Ijl 

,50 

.51 

28 

206 

153 

120 

99 

83 

72 

83 

57 

51 

47 

43 

30 

199 

148 

115 

94 

79 

68 

M 

53 

48 

43 

40 

33 

35 

188 

137 

106 

85 

71 

60 

.52 

46 

41 

37 

33 

27 

23 

40 

179 

129 

99 

79 

65 

54 

47 

41 

36 

32 

29 

23 

!',< 

16 

45 

172 

124 

93 

74 

60 

50 

43 

37 

32 

M 

2.5 

20 

16 

13 

11 

50 

167 

119 

89 

70 

57 

47 

39 

34 

29 

26 

23 

18 

14 

11 

I 

8 

!  55 

162 

115 

86 

67 

54 

44 

37 

32 

27 

24 

21 

16 

12 

10 

8 

7 

5 

60 

159 

112 

83 

64 

51 

42 

35 

M 

25 

22 

19 

14 

11 

9 

7 

5 

4 

4 

65 

155 

109 

80 

62 

49 

40 

3:.l 

2s 

24 

21 

18 

13 

10 

7 

6 

t 

4 

3 

2 

70 

152 

106 

78 

60 

48 

38 

32 

27 

23 

19 

17 

12 

9 

7 

5 

4 

3 

2 

i 

1 

75 

150 

104 

76 

58 

46 

37 

30 

25 

21 

u 

16 

11 

S 

6 

4 

g 

2 

2 

i 

1 

1 

80 

147 

102 

74 

57 

45 

36 

29 

24 

2(1 

17 

1.5 

10 

7 

.5 

4 

3 

2 

1 

i 

1 

0 

0 

85 

145 

100 

73 

55 

"43 

34 

M 

23 

U 

16 

14 

10 

7 

,5 

3 

l" 

*2 

1 

i 

0 

0 

0 

d 

90 

143 

98 

71 

54 

42 

33 

27 

22 

u 

16 

13 

g 

6 

4 

3 

2 

1 

1 

i 

0 

u 

0 

0 

95 

140 

96 

70 

53 

41 

32 

M 

22 

is 

1.5 

13 

y 

1 

4 

3 

2 

1 

0 

0 

0 

0 

100 

138 

95 

68 

51 

40 

31 

25 

21 

17 

14 

12 

8 

6 

4 

3 

2 

1 

0 

0 

0 

105 

136 

93 

67 

50 

39 

30 

25 

20 

17 

14 

12 

8 

,5 

4 

2 

2 

1 

0 

0 

110 

134 

91 

65 

49 

38 

30 

24 

1!> 

16 

13 

11 

7 

.5 

3 

2 

2 

1 

1 

115 

132 

89 

64 

48 

37 

29 

23 

1!. 

1.5 

13 

11 

7 

.5 

3 

2 

2 

1 

120 

129 

88 

62 

46 

36 

28 

22 

is 

1.5 

12 

10 

7 

.5 

3 

2 

2 

125 

127 

86 

61 

45 

35 

27 

22 

u 

14 

12 

10 

7 

.5 

4 

3 

2 

130 

125 

84 

59 

44 

34 

26 

21 

17 

14 

12 

10 

7 

5 

4 

3 

135 

122 

82 

58 

43 

33 

26 

21 

17 

14 

12 

10 

7 

.5 

4 

140 

119 

80 

56 

42 

32 

25 

2(1 

17 

14 

12 

10 

8 

6 

145 

116 

77 

55 

41 

32 

25 

21 

17 

1.5 

13 

11 

g 

150 

112 

75 

54 

40 

32 

26 

21 

U 

16 

U 

13 

152 

111 

75 

53 

40 

32 

26 

22 

1" 

17 

16 

154 

110 

74 

53 

41 

33 

27 

23 

21 

U 

156 

108 

74 

54 

42 

34 

28 

2.5 

22 

158 

107 

74 

54 

43 

35 

30 

27 

160 

107 

74 

56 

45 

38 

33 

162 

107 

76 

59 

48 

42 

164 

109 

'  79 

63 

54 

166 

113 

86 

71 

168 

122 

98 

170 

143 

334 


TABLES 


TABLE  II.  —  CORRECTION  FOR  EARTH'S  CURVATURE  AND 

REFRACTION 


Dist. 

Corr. 

Dist. 

Corr. 

Dist. 

Corr. 

Miles. 

1 

Feet. 
0.6 

Miles. 
21 

Feet. 
253.1 

Miles. 
41 

Feet. 
964.7 

2 

2.3 

22 

277.7 

42 

1012.2 

3 

5.2 

23 

303.6 

43 

1061.0 

4 

9.2 

24 

330.5 

44 

1111.0 

5 

14.4 

25 

358.6 

45 

1162.0 

6 

20.6 

26 

388.0 

46 

1214.2 

7 

28.1 

27 

418.3 

47 

1267.7 

8 

36.7 

28 

449.9 

48 

1322.1 

9 

46.4 

29 

482.6 

49 

1377.7 

10 

57.4 

30 

516.4 

50      ' 

1434.6 

11 

69.4 

31 

551.4 

51 

1492.5 

12 

82.7 

32 

587.6 

52 

1551.6 

13 

97.0 

33 

624.9 

53 

1611.9 

14 

112.5 

34 

663.3 

54 

1673.3 

15 

129.1 

35 

703.0 

55 

1735.8 

16 

146.9 

36 

743.7 

56  . 

1799.6 

17 

165.8 

37 

785.6 

57 

1864.4 

18 

185.9 

38 

828.6 

58 

1930.4 

19 

207.2 

39 

872.8 

59 

1997.5 

20 

229.5 

40 

918.1 

60 

2065.8 

TABLES 


335 


TABLE    III.  — SHORT    TABLE    OF    FACTORS    FOR    REDUCTION 
OF  TRANSIT  OBSERVATIONS 

Top  Argument  =  Star's  Declination  (5). 
•  Side  Argument  =  Star's  Zenith  Distance  (f) . 

[For  factor  A  use  left-hand  argument.    For  factor  B  use  right-hand  argument.    For  factor  C 
use  bottom  line.J 


r 

0° 

Ifr 

15° 

20° 

25° 

30° 

35° 

40° 

45° 

50° 

55° 

60° 

65° 

70° 

r 

1° 

5 
10 

0.02 
0.09 
p.  17 

0.02 
0.09 
0.18 

0.02 
0.09 
0.18 

0.02 
0.09 
0.19 

0.02 
0.10 
0.19 

0.02 
0.10 
0.20 

0.02 
0.11 
0.21 

0.02 
0.11 
0.23 

0.02 
0.12 
0.25 

0.03 
0.13 
0.27 

0.03 
0.15 
0.30 

0.03 
0.17 
0.35 

0.04 
0.21 
0.41 

0.05 
0.25 
0.51 

89° 
85 
80 

15 
20 
25 

0.26 
0.34 
0.42 

0.26 
0.35 
0.43 

0.27 
0.35 
0.44 

0.28 
0.36 
0.45 

0.29 
0.38 
0.47 

0.30 
0.40 
0.49 

0.32 
0.42 
0.52 

0.34 
0.45 
0.55 

0.37 
0.48 
0.60 

0.40 
0.53 
0.66 

0.45 
0.60 
0.74 

0.52 
0.68 
0.85 

0.61 
0.81 
1.00 

0.76 
1.00 
1.24 

75 
70 
65 

30 

35 
40 

0.50 
0.57 
0.64 

0.51 
0.58 
0.65 

0.52 
0.59 
0.67 

0.53 
0.61 
0.68 

0.55 
0.63 
0.71 

0.58 
0.66 
0.74 

0.61 
0.70 

0.78 

0.65 
0.75 
0.84 

0.71 
0.81 
0.91 

0.78 
0.89 
1.00 

0.87 
1.00 
1.12 

1.00 
1.15 
1.29 

1.18 
.36 
.52 

1.46 
1.68 

1.88 

60 
55 
50 

45 
50 
55 

0.71 
0.77 
0.82 

0.72 
0.78 
0.83 

0.73 
0.79 
0.85 

0.75 
0.82 
0.87 

0.78 
0.85 
0.90 

0.82 
0.89 
0.95 

0.86 
0.94 
1.00 

0.92 
1.00 
1.07 

1.00 
1.08 
1.16 

1.10 
1.19 
1.27 

.23 
.34 
.43 

1.41 
1.53 
1.64 

.67 
.81 
.94 

2.07 
2.24 
2.40 

45 
40 
35 

60 
65 
70 

0.87 
0.91 
0.94 

0.88 
0.92 
0.95 

0.90 
0.94 
0.97 

0.92 
0.96 
1.00 

0.96 
1.00 
1.04 

.00 
.05 
.09 

1.06 
1.11 
1.15 

1.13 
1.18 
1.23 

.22 
.28 
.33 

1.35 
1.41 
1.46 

.51 

.58 
.64 

1.73 
1.81 

1.88 

2.05 
2.14 
2.22 

2.53 
2.65 
2.75 

30 
25 
20 

75 

80 

85 

0.97 
0.98 
1.00 

0.98 
1.00 
1.01 

1.00 
1.02 
1.03 

1.03 
1.05 
1.06 

.07 
.09 
.10 

.12 
.14 
.15 

1.18 
1.20 
1.22 

1.26 
1.29 
1.30 

.37 
.39 
.41 

1.50 
1.53 
1.55 

1.68 
1.72 
1.74 

1.93 
1.97 
1.99 

2.29 
2.33 
2.36 

2.82 
2.88 
2.91 

15 
10 
5 

90 

1.00 

1.02 

1.04 

1.06 

1.10 

1.15 

1.22 

1.31 

1.41 

1.56 

1.74 

2.00 

2.37 

2.92 

0 

TABLE  IV. —DIURNAL  ABERRATION   (*c) 


Lati- 

Declination  =  5. 

=  <*>• 

0° 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

75° 

80° 

85° 

0 

0.02 

0.02 

0.02 

0*02 

0*03 

0*03 

0*04 

0*06 

0*08 

0*12 

0.24 

10 

0.02 

0.02 

0.02 

0.02 

0.03 

0.03 

0.04 

0.06 

0.08 

0.12 

0.24 

20 

0.02 

0.02 

0.02 

0.02 

0.03 

0.03 

0.04 

0.06 

0.08 

0.11 

0.23 

30 

0.02 

0.02 

0.02 

0.02 

0.02 

0.03 

0.04 

0.05 

0.07 

0.10 

0.21 

40 

0.02 

0.02 

0.02 

0.02 

0.02 

0.03 

0.03 

0.05 

0.06 

0.09 

0.18 

50 

0.01 

0.01 

0.01 

0.02 

0.02 

0.02 

0.03 

0.04 

0.05 

0.08 

0.15 

60 

0.01 

0.01 

0.01 

0.01 

0.01 

0.02 

0.02 

0.03 

0.04 

0  06 

0.12 

70 

0.01 

0.01 

0.01 

•0.01 

0.01 

0.01 

0.01 

0.02 

0.03 

0.04 

0.08 

80 

0.00 

0.00 

0.00 

0.00 

0.00 

0.01 

0.01 

0.01 

0.01 

0.02 

0.04 

336 


TABLES 


TABLE  V.  —  CORRECTION  TO  LATITUDE  FOR  DIFFEREN 
TIAL  REFRACTION  =  %  (r  -  r'). 
(The  sign  of  the  correction  is  the  same  as  that  of  the  micrometer  difference.] 


One-half 
diff.of 
zenith 
distances. 

Zenith  distance. 

0° 

10° 

20° 

25° 

30° 

35° 

40° 

45° 

0.0 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.5 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

0.02    , 

1.0 

0.02 

0.02 

0.02 

0.02 

0.02 

0.03 

0.03 

0.03 

1.5 

0.03 

0.03 

0.03 

0.03 

0.03 

0.04 

0.04 

0.05 

2.0 

0.03 

0.03 

0.04 

0.04 

0.04 

0.05 

0.06 

0.07 

2.5 

0.04 

0.04 

0.05 

0.05 

0.06 

0.06 

0.07 

0.08 

3.0 

0.05 

0.05 

0.06 

0.06 

0.07 

0.08 

0.09 

0.10 

3.5 

0.06 

0.06 

0.07 

0.07 

0.08 

0.09 

0.10 

0.12 

4.0 

0.07 

0.07 

0.08 

0.08 

0.09 

0.10 

0.11 

0.13 

4.5 

0.08 

0.08 

0.09 

0.09 

0.10 

0.11 

0.13 

0.15 

5.0 

0.08 

0.09 

0.10 

0.10 

0.11 

0.13 

0.14 

0.17 

5.5 

0.09 

0.10 

0.10 

0.11 

0.12 

0.14 

0.16 

0.18 

6.0 

0.10 

0.10 

0.11 

0.12 

0.13 

0.15 

0.17 

0.20 

6.5 

0.11 

0.11 

0.12 

0.13 

0.14 

0.16 

0.19 

0.22 

7.0 

0.12 

0.12 

0.13 

0.14 

0.16 

0.18 

0.20 

0.23 

7.5 

0.13 

0.13 

0.14 

0.15 

0.17 

0.19 

0.21 

0.25 

8.0 

0.13 

0.14 

0.15 

0.16 

0.18 

0.20 

0.23 

0.27 

8.5 

0.14 

0.15 

0.16 

0.17 

0.19 

0.21 

0.24 

0.29 

9.0 

0.15 

0.16 

0.17 

0.18 

0.20 

0.23 

0.26 

0.30 

9.5 

0.16 

0.16 

0.18 

0.19 

0.21 

0.24 

0.27 

0.32 

10.0 

0.17 

0.17 

0.19 

0.20 

0.22 

0.25 

0.29 

0.34 

10.5 

0.18 

0.18 

0.20 

0.21 

0.23 

0.26 

0.30 

0.35 

11.0 

0.18 

0.19 

0.21 

0.22 

0.25 

0.28 

0.31 

0.37 

11.5 

0.19 

0.20 

0.22 

0.23 

0.26 

0.29 

0.33 

0.39 

12.0 

0.20 

0.21 

0.23 

0.25 

0.27 

0.30 

0.34 

0.40 

12.5 

0.21 

0.22 

0.24 

0.26 

0.28 

0.31 

0.36 

0.42 

13.0 

0.22 

0.22 

0.25 

0.27 

0.29 

0.33 

0.37 

0.44 

13.5 

0.23 

0.23 

0.26 

0.28 

0.30 

0.34 

0.39 

0.45 

14.0 

0.23 

0.24 

0.27 

0.29 

0.31 

0.35 

0.40 

0.47 

14.5 

0.24 

0.25 

0.28 

0.30 

0.32 

0.36 

0.41 

0.49 

15.0 

0.25 

0.26 

0.29 

0.31 

0.34 

0.38 

0.43 

0.50 

15.5 

0.26 

0.27 

0.29 

0.32 

0.35 

0.39 

0.44 

0.52 

16.0 

0.27 

0.28 

0.30 

0.33 

0.36 

0.40 

0.46 

0.54 

16.5 

0.28 

0.29 

0.31 

0.34 

0.37 

0.41 

0.47 

0.55 

17.0 

0.29 

0.29 

0.32 

0.35 

0.38 

0.43 

0.49 

0.57 

17.5 

0.29 

0.30 

0.33 

0.36 

0.39 

0.44 

0.50 

0.59 

18.0 

0.30 

0.31 

0.34 

0.37 

0.40 

0.45 

0.51 

0.60 

18.5 

0.31 

0.32 

0.35 

0.38 

0.41 

0.46 

0.53 

0.62 

19.0 

0.32 

0.33 

0.36 

0.39 

0.43 

0.48 

0.54 

0.64 

19.5 

0.33 

0.34 

0.37 

0.40 

0.44 

0.49 

0.56 

0.65 

20.0 

0.34 

0.35 

0.38 

0.41 

0.45 

0.50 

0.57 

0.67 

TABLES 


337 


TABLE    VI.  —  CORRECTION    TO    LATITUDE    FOR    REDUCTION 

TO  MERIDIAN 

[Star  off  the  meridian  but  instrument  in  the  meridian.    The  sign  of  the  correction  to  the 
latitude  is  positive  except  for  stars^sputh  of  the  equator  and  subpolars.] 


s 

10« 

15« 

20» 

22* 

24« 

26* 

28« 

30* 

' 

!2* 

3- 

• 

36* 

38* 

S 

1 

" 

" 

" 

" 

i 
0. 

t—  • 

01 

0.01 

0.01 

89 

2 

0.01 

0.01 

0.01 

0.01 

0 

.01 

0. 

01 

0.01 

0.01 

88 

3 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

(I 

.01 

0. 

a 

0.02 

0.02 

87 

4 

0.01 

0.01 

0:01 

0.01 

0.01 

0.02 

0 

.02 

0. 

)2 

0.02 

0.03 

86 

5 

0.01 

0.01 

0.01 

0.01 

0.02 

0.02 

0.02 

0 

.02 

o. 

a 

0.03 

0.03 

85 

6 

0.01 

0.01 

0.01 

0.02 

0.02 

0.02 

0.03 

0 

.03 

0. 

• 

0.04 

0.04 

84 

7 

o.oi 

0.01 

0.02 

0.02 

0.02 

0.03 

0.03 

0 

.03 

0. 

J4 

0.04 

0.05 

83 

8 

0.01 

0.02 

0.02 

0.02 

0.03 

0.03 

0.03 

0 

.04 

0. 

M 

0.05 

0.05 

82 

9 

0.01 

0.02 

0.02 

0.02 

0.03 

0.03 

0.04 

0 

.04 

0. 

0.5 

0.05 

0.06 

81 

10 

0.01 

0.02 

0.02 

0.03 

0.03 

0.04 

0.04 

0 

.05 

0. 

M 

0.06 

0.07 

80 

12 

0.01 

0.01 

0.02 

0.03 

0.03 

0.04 

0.05 

0.05 

0 

.06 

0. 

06 

0.07 

0.08 

78 

14 

0.01 

0.01 

0.03 

0.03 

0.04 

0.04 

0.05 

0.06 

0 

.07 

0. 

07 

0.08 

0.09 

76 

16 

0.01 

ft.  02 

0.03 

0.03 

0.04 

0.05 

0.06 

0.07 

0 

.07 

n. 

M 

0.09 

0.10 

74 

18 

0.01 

0.02 

0.03 

0.04 

0.05 

0.05 

0.06 

0.07 

0 

.08 

D. 

JO 

0.10 

0.12 

72 

20 

0.01 

0.02 

0.04 

0  04 

0.05 

0.06 

0.07 

0.08 

0 

.09 

0. 

10 

0.11 

0.13 

70 

22 

0.01 

0.02 

0.04 

0.05 

0.05 

0.06 

0.07 

0.09 

0 

.10 

0. 

11 

0.12 

0.14 

68 

24 

0.01 

0.02 

0.04 

0.05 

0.06 

0.07 

0.08 

0.09 

0 

.10 

0. 

12 

0.13 

0.15 

66 

26 

0.01 

0.02 

0.04 

0.05 

0.06 

0.07 

0.08 

0.10 

0 

.11 

0. 

12 

0.14 

0.15 

64 

28 

0.01 

0.03 

0.05 

0.05 

0.07 

0.08 

0.09 

0.10 

0 

:12 

0. 

13 

0.15 

0.16 

62 

30 

0.01 

0.03 

0.05 

0.06 

0.07 

0.08 

0.09 

0.11 

0 

.12 

0. 

14 

0.15 

0.17 

60 

32 

0.01 

0.03 

0.05 

0.06 

0.07 

0.08 

0.10 

0.11 

0 

.13 

0. 

14 

0.16 

0.18 

58 

34 

•  0.01 

0.03 

0.05 

0.06 

0.07 

0.09 

0.10 

0.11 

0 

.13 

0. 

15 

0.16 

0.18 

56 

36 

0.01 

0.03 

0.05 

0.06 

0.07 

0.09 

0.10 

0.12 

0 

.13 

0. 

1.5 

0.17 

0.19 

54 

38 

0.01 

0.03 

0.05 

0.06 

0.08 

0.09 

0.10 

0.12 

0 

13 

0. 

1.5 

0.17 

0.19 

52 

40 

0.01 

0.03 

0.05 

0-07 

0.08 

0.09 

0.11 

0.12 

0 

14 

0 

16 

0.17 

0.19 

50 

45 

0.01 

0.03 

0.05 

0.07 

0.08 

0.09 

0.11 

0.12 

0 

14 

0 

16 

0.18 

0.20 

45 

S 

40* 

42* 

44* 

46* 

.48* 

50* 

52* 

54< 

r 

5( 

« 

58* 

60* 

d 

1 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

0.( 

»i 

0 

01 

).02 

0.02 

89 

2 

0.02 

0.02 

0.02 

0.02 

0.02 

0.02 

0.03 

0.( 

1 

0 

09 

i 

).03 

0.03 

88 

3 

iO.02 

0.03 

0.03 

0.03 

0.03 

0.04 

0.04 

0.( 

4 

0 

04 

1 

).05 

0.05 

87 

4 

0.03 

0.03 

0.04 

0.04 

0.04 

0.05 

0.05 

0.( 

8 

0 

M 

• 

).06 

0.07 

86 

5 

0.04 

0.04 

0.05 

0.05 

0.05 

0.06 

0.06 

0.( 

7 

0 

07 

1 

).08 

0.09 

85 

6 

0.05 

0.05 

0.06 

0  06 

0.07 

0.07 

0.08 

0.( 

I 

0 

bo 

( 

UO 

0.10 

84 

7 

0.05 

0.06 

0.06 

0.07 

0.08 

0.08 

0.09 

0.1 

0 

0 

10 

i 

>.ll 

0.12 

83 

8 

0.06 

0.07 

0.07 

0.08 

0.09 

0.09 

0.10 

OJ 

1 

0 

12 

i 

1.13 

0.14 

82 

9 

0.07 

0.07 

0.08 

0.09 

0.10 

0.11 

0.11 

0.1 

2 

0. 

13 

( 

.14 

0.15 

81 

10 

0.07 

0.08 

0.09 

0.10 

0.11 

0.12 

0.13 

0.1 

4 

0. 

1,5 

( 

.16 

0.17 

80 

12 

0.09 

0.10 

0.11 

0.12 

0.13 

0.14 

0.15 

0.1 

1 

0. 

17 

t 

.19 

0.20 

78 

14 

0.10 

0.11 

0.12 

0.14 

0.15 

0.16 

0.17 

O.I 

'.i 

0. 

20 

I 

.22 

0.23 

76 

16 

0.12 

0.13 

0.14 

0.15 

0.17 

0.18 

0.20 

0.2 

1 

0. 

23 

I 

.24 

0.26 

74 

18 

0.13 

0.14 

0.16 

0.17 

0.18 

0.20 

0.22 

0.2 

I 

0. 

2.5 

C 

.27 

0.29 

72 

20 

0.14 

0.15 

0.17 

0.19 

0.20 

0.22 

0.24 

0.2 

1 

0. 

28 

'  C 

.29 

0.32 

70 

22 

0.15 

0.17 

Q.  18 

0.20 

0.22 

0.24 

0.26 

0.2 

B 

0. 

30 

0 

.32 

0.34 

68 

24 

0.16 

0.18 

0.20 

0.21 

0.23 

0.25 

0.27 

0.2 

9 

0. 

H 

1 

.34 

0  36 

66 

26 

0.17 

0.19 

0.21 

0.23 

0.25 

0.27 

0.29 

0.3 

1 

0. 

U 

1 

.36 

0.39 

64 

28 

0.18 

0.20 

0.22 

0.24 

0.26 

0.28 

0.31 

0.3 

3 

0. 

IS 

n 

.38 

0.41 

62 

30 

0.19 

0.21 

0.23 

0.25 

0.27 

0.30 

0.32 

0.3 

I 

0. 

n 

r 

.40 

0.42 

60 

32 

0.20 

0.22 

0.24 

0.26 

0.28 

0.31 

0.33 

0.3 

| 

0. 

3ft 

0 

.41 

0.44 

58 

34 

0.20 

0.22 

0.24 

0.27 

0.29 

0.32 

0.34 

0.3 

; 

0. 

40 

0 

.42 

0.45 

56 

36 

0.21 

0.23 

0.25 

0.28 

0.30 

0.32 

0.35 

0.3 

s 

0. 

41 

0 

.44 

0.47 

54 

38 

0.21 

0.23 

0.26 

0.28 

0  30 

0.33 

0.36 

0.3 

» 

0. 

41 

0 

.44 

0.48 

52 

40 

0.21 

0.24 

0.26 

0.28 

0.31 

0.34 

0.36 

0.3 

• 

0. 

42 

0 

45 

0.48 

50 

45 

0.22 

0.24 

0.26 

0.29 

0.31 

0.34 

0.37 

0.4 

i 

0. 

43 

0 

.46 

0.49 

45 

338 


TABLES 


TABLE  VII.  — REDUCTION   OF  LATITUDE  TO   SEA  LEVEL 

[The  correction  is  negative  in  every  case.] 


5° 

85° 

10° 
80° 

15° 

75° 

20° 
70° 

25° 
65° 

30° 
60° 

35° 
55° 

40° 
50° 

45° 

Feet. 
100 

Meters 

30 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.01 

0.01 

200 

61 

0.00 

0.00 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

300 

91 

0.00 

0.01 

0.01 

0.01 

0.01 

0.01 

0.01 

0.02 

0.02 

400 

122 

0.00 

0.01 

0.01 

0.01 

0.02 

0.02 

0.02 

0.02 

0.02 

500 

152 

0.00 

0.01 

0.01 

0.02 

0.02 

0.02 

0.02 

0.03 

0.03 

600 

183 

0.01 

0.01 

0.02 

0.02 

0.02 

0.03 

0.03 

0.03 

0.03 

700 

213 

0.01 

0.01 

0.02 

0.02 

0.03 

0.03 

0.03 

0.04 

0.04 

800 

244 

0.01 

0.01 

0.02 

0.03 

0.03 

0.04 

0.04 

0.04 

0.04 

900 

274 

0.01 

0.02 

0.02 

0.03 

0.04 

0.04 

0.04 

•0.05 

0.05 

1000 

305 

0.01 

0.02 

0.03 

0.03 

0.04 

0.05 

0.05 

0.05 

0.05 

1100 

335 

0.01 

0.02 

0.03 

0.04 

0.04 

0.05 

0.05 

0.06 

0.06 

1200 

366 

0,01 

0.02 

0.03 

0.04 

0.05 

0.05 

0.06 

0.06 

0.06 

1300 

396 

0.01 

0.02 

0.03 

0.04 

0.05 

0.06 

0.06 

0.07 

0.07 

1400 

427 

0.01 

0.02 

0.04 

0.05 

0.06 

0.06 

0.07 

0.07 

0.07 

1500 

457 

0.01 

0.03 

0.04 

0.05 

0.06 

0.07 

0.07 

0.08 

0.08 

1600 

488 

0.01 

0.03 

0.04 

0.05 

0.06 

0.07 

0.08 

0.08 

0.08 

1700 

518 

0.02 

0.03 

0.04 

0.06 

0.07 

0.08 

0.08 

0.09 

0.09 

1800 

549 

0.02 

0.03 

0.05 

0.06 

0.07 

0.08 

0.09 

0.09 

0.09 

1900 

579 

0.02 

0.03 

0.05 

0.06 

0.08 

0.09 

0.09 

0.10 

0.10 

2000 

610 

0.02 

0.04 

0.05 

0.07 

0.08 

0.09 

0.10 

0.10 

0.10 

2100 

640 

0.02 

0.04 

0.05 

0.07 

0.08 

0.09 

0.10 

0.11 

0.11 

2200 

671 

0.02 

0.04 

0.06 

0.07 

0.09 

0.10 

0.11 

0.11 

0.11 

2300 

701 

0.02 

0.04 

0.06 

0.08 

0.09 

0.10 

0.11 

0.12 

0.12 

2400 

732 

0.02 

0.04 

0.06 

0.08 

0.10 

0.11 

0.12 

0.12 

0.13 

2500 

762 

0.02 

0.04 

0.07 

0.08 

0.10 

0.11 

0.12 

0.13 

0.13 

2600 

792 

0.02 

0.05 

0.07 

0.09 

0.10 

0.12 

0.13 

0.13 

0.14 

2700 

823 

0.02 

0.05 

0.07 

0.09 

0.11 

0.12 

0.13 

0.14 

0.14 

2800 

853 

0.03 

0.05 

0.07 

0.09 

0.11 

0.13 

0.14 

0.14 

0.15 

2900 

884 

0.03 

0.05 

0.08 

0.10 

0.12 

0.13 

0.14 

0.15 

0.15 

3000 

914 

0.03 

0.05 

0.08 

0.10 

0.12 

0.14 

0.15 

0.15 

0.16 

3100 

945 

0.03 

0.06 

0.08 

0.10 

0.12 

0.14 

0.15 

0.16 

0.16 

3200 

975 

0.03 

0.06 

0.08 

0.11 

0.13 

0.14 

0.16 

0.16 

0.17 

3300 

1006 

0.03 

0.06 

0.09 

0.11 

0.13 

0.15 

0.16 

0.17 

0.17 

3400 

1036 

0.03 

0.06 

0.09 

0.11 

0.12 

0.15 

0.17 

0.17 

0.18 

i  3500 

1067 

0.03 

0.06 

0.09 

0.12 

0.14 

0.16 

0.17 

0.18 

0.18 

TABLES 


339 


TABLE  VII  (Con.).  — REDUCTION  OF  LATITUDE  TO  SEA  LEVEL 

(The  correction  is  negative  in  every  case.] 


5° 

85° 

10° 
80° 

15° 
75° 

20° 
70° 

25° 
65° 

30° 
60° 

35° 
55° 

40° 
50° 

45° 

Feet. 
3600 

Meters. 
1097 

0.03 

0.06 

0.09 

0.12 

0.14 

0.16 

0.18 

0.18 

0.19 

3700 

1128 

0.03 

0.07 

0.10 

0.12 

0.15 

0.17 

0.18 

0.19 

0.19 

3800 

1158 

0.03 

0.07 

0.10 

0.13 

0.15 

0.17 

0.19 

0.20 

0.20 

3900 

1189 

0.04 

0.07 

0.10 

0.13 

0.16 

0.18 

0.19 

0.20 

0.20 

4000 

1219 

0.04 

0.07 

0.10 

0.13 

0.16 

0.18 

0.20 

0.21 

0.21 

4100 

1250 

0.04 

0.07 

0.11 

0.14 

0.16 

0.19 

0.20 

0.21 

0.21 

4200 

1280 

0.04 

0.07 

0.11 

0.14 

0.17 

0.19 

0.21 

0.22 

0.22 

4300 

1311 

0.04 

0.08 

0.11 

0.14 

0.17 

0.19 

0.21 

0.22 

0.22 

4400 

1341 

0.04 

0.08 

0.11 

0.15 

0.18 

0.20 

0.22 

0.23 

0.23 

4500 

1372 

0.04 

0.08 

0.12 

0.15 

0.18 

0.20 

0.22 

0.23 

0.23 

4600 

1402 

0.04 

0.08 

0.12 

0.15 

0.18 

0.21 

0.23 

0.24 

0.24 

4700 

1433 

0.04 

0.08 

0.12 

0.16 

0.19 

0.21 

0.23 

0.24 

0.24 

4800 

1463 

0.04 

0.09 

0.13 

0.16 

0.19 

0.22 

0.24 

0.25 

0.25 

4900 

1494 

0.04 

0.09 

0.13 

0.16 

0.20 

0.22 

0.24 

0.25 

0.26 

5000 

1524 

0.05 

0.09 

0.13 

0.17 

0.20 

0.23 

0.24 

0.26 

0.26 

5100 

1554 

0.05 

0.09 

0.13 

0.17 

0.20 

0.23 

0.25 

©.26 

0.27 

5200 

1585 

0.05 

0.09 

0.14 

0.17 

0.21 

0.23 

0.25 

0.27 

0.27 

5300 

1615 

0.05 

0.09 

0.14 

0.18 

0.21 

0.24 

0.26 

0.27 

0.28 

5400 

1646 

0.05 

0.10 

0.14 

0.18 

0.22 

0.24 

0.26 

0.28 

0.28 

5500 

1676 

0.05 

0.10 

0.14 

0.18 

0.22 

0.25 

0.27 

0.28 

0.29 

5600 

1707 

0.05 

0.10 

0.15 

0.19 

0.22 

0.25 

0.27 

0.29 

0.29 

5700 

1737 

0.05 

0.10 

0.15 

0.19 

0.23 

0.26 

0.28 

0.29 

0.30 

5800 

1768 

0.05 

0.10 

0.15 

0.19 

0.23 

0.26 

0.28 

0.30 

0.30 

5900 

1798 

0.05 

0.11 

0.15 

0.20 

0.24 

0.27 

0.29 

0.30 

0.31 

6000 

1829 

0.05 

0.11 

0.16 

0.20 

0.24 

0.27 

0.29 

0.31 

0.31 

6100 

1859 

0.06 

0.11 

0.16 

0.20 

0.24 

0.28 

G.30 

0.31 

0.32 

6200 

1890 

0.06 

0.11 

0.16 

0.21 

0.25 

0.28 

0.30 

0.32 

0.32 

6300 

1920 

0.06 

0.11 

0.16 

0.21 

0.25 

0.28 

0.31 

0.32 

0.33 

6400 

1951 

0.06 

0.11 

0.17 

0.21 

0.26 

0.29 

0.31 

0.33 

0.33 

6500 

1981 

0.06 

0.12 

0.17 

0.22 

0.26 

0.29 

0.32 

0.33 

0.34 

6600 

2012 

0.06 

0.12 

0.17 

0.22 

0.26 

0.30 

0.32 

0.34 

0.34 

6700 

2042 

0.06 

0.12 

0.17 

0.22 

0.27 

0.30 

0.33 

0.34 

0.35 

6800 

2073 

0.06 

0.12 

0.18 

0.23 

0.27 

0.31 

0.33 

0.35 

0.35 

6900 

2103 

0.06 

0.12 

0.18 

0.23 

0.28 

0.31 

0.34 

0.35 

0.36 

7000 

2134 

0.06 

0.12 

0.18 

0.23 

0.28 

0.32 

0.34 

0.36 

0.36 

340 


TABLES 


TABLE  VII  (Con.).  — REDUCTION  OF  LATITUDE  TO  SEA  LEVEL 

[The  correction  is  negative  in  every  case.) 


X.      <6 

*  X 

5° 
85° 

10° 
80° 

15° 

75° 

20° 
70° 

25° 
65° 

30° 
60° 

35° 
55° 

40° 
50° 

45° 

Feet. 

Meters. 

» 

» 

// 

; 

» 

// 

// 

" 

n 

7100 

2164 

0.06 

0.13 

0.19 

0.24 

0.28 

0.32 

0.35 

0.36 

0.37 

7200 

2195 

0.07 

0.13 

0.19 

0.24 

0.29 

0.33 

0.35 

0.37 

0.38 

7300 

2225 

0.07 

0.13 

0.19 

0  24 

0.29 

0.33 

0.36 

0.37 

0.38 

7400 

2256 

0.07 

0.13 

0.19 

0  25 

0.30 

0.33 

0.36 

0.38 

0.39 

7500 

2286 

0.07 

0.13 

0.20 

0.25 

0.30 

0.34 

0.37 

0.38 

0.39 

7600 

2316 

0.07 

0.14 

0.20 

0.25 

0.30 

0.34 

0.37 

0.39 

0.40 

7700 

2347 

0.07 

0.14 

0.20 

0.26 

0.31 

0.35 

0.38 

0.40 

0.40 

7800 

2377 

0.07 

0.14 

0.20 

0.26 

0.31 

0.35 

0.38 

0.40 

0.41 

7900 

2408 

0.07 

0.14 

0.21 

0.26 

0.32 

0.36 

0.39 

0.41 

0.41 

8000 

2438 

0.07 

0.14 

0.21 

0.27 

0.32 

0.36 

0.39 

0.41 

0.42 

8100 

2469 

0.07 

0.14 

0.21 

0.27 

0.32 

0.37 

0.40 

0.42 

0.42 

8200 

2499 

0.07 

0.15 

0.21 

0.27 

0.33 

0.37 

0.40 

0.42 

0.43 

8300 

2530 

0.08 

0.15 

0.22 

0.28 

0.33 

0.37 

0.41 

0.43 

0.43 

8400 

2560 

0.08 

0.15 

0.22 

0.28 

0.34 

0.38 

0.41 

0.43 

0.44 

8500 

2591 

0.08 

0.15 

0.22 

0.28 

0.34 

0.38 

0.42 

0.44 

0.44 

<600 

2621 

0.08 

0.15 

0.22 

0.29 

0.34 

0.39 

0.42 

0.44 

0.45 

8700 

2652 

0.08 

0.16 

0.23 

0.29 

0.35 

0.39 

0.43 

0.45 

0.45 

8800 

2682 

0.08 

0.16 

0.23 

0.29 

0.35 

0.40 

0.43 

0.45 

0.46 

8900 

2713 

0.08 

0.16 

0.23 

0.30 

0.36 

0.40 

0.44 

0.46 

0.46 

9000 

2743 

0.08 

0.16 

0.23 

0.30 

0.36 

0.41 

0.44 

0.46 

0.47 

9100 

2774 

0.08 

0.16 

0.24 

0.30 

0.36 

0.41 

0.45 

0.47 

0.47 

9200 

2804 

0.08 

0.16 

0.24 

0.31 

0.37 

0.42 

0.45 

0.47 

0.48 

9300 

2835 

0.08 

0.17 

0.24 

0.31 

0.37 

0.42 

0.46 

0.48 

0.48 

9400 

2865 

0.09 

0.17 

0.24 

0.31 

0.38 

0.42 

0.46 

0.48 

0.49 

9500 

2896 

0.09 

0.17 

0.25 

0.32 

0.38 

0.43 

0.47 

0.49 

0.50 

9600 

2926 

0.09 

0.17 

0.25 

0.32 

0.38 

0.43 

0.47 

0.49 

0.50 

9700 

2957 

0.09 

0.17 

0.25 

0.32 

0.39 

0.44 

0.48 

0.50 

0.51 

9800 

2987 

0.09 

0.17 

0.26. 

0.33 

0.39 

0.44 

0.48 

0.50 

0.51 

9900 

3018 

0.09 

0.18 

0.26 

0.33 

0.40 

0.45 

0.48 

0.51 

0.52 

10000 

3048 

0.09 

0.18 

0.26 

0.33 

0.40 

0.45 

0.49 

0.51 

0.52 

TABLES 


341 


TABLE  VIII.  — FOR  CONVERTING  SIDEREAL  INTO  MEAN 
SOLAR  TIME 

[Increase  in  Sun's  Right  Ascension  in  Sidereal  h.  m.  s.] 

Mean  Time  =  Sidereal  Time  —  C. 


Sid. 
Hrs. 

Corr. 

Sid. 
Min. 

Corr. 

Sid. 
Min. 

Corr. 

Sid. 
Sec. 

Corr. 

Sid. 
Sec. 

Corr. 

I 

m         s 
o     9  .830 

I 

s 
0.164 

31 

s 
5-079 

I 

s 
0.003 

31 

s 
0.085 

2 

o  19  .659 

2 

0.328 

32 

5.242 

2 

0.005 

32 

0.087 

3 

o  29  .489 

3 

0.491 

33 

5.406 

3 

o  .008 

33 

o  .090 

4 

o  39.318 

4 

0.655 

34 

5-57° 

4 

o  .on 

34 

0.093 

5 

o  49  .148 

5 

o  .819 

35 

5-734 

5 

0.014 

35 

0.096 

6 

o  58  .977 

6 

0.983 

36 

5.898 

6 

0.016 

36 

0.098 

I 

i     8.807 
i  18.636 

I 

.147 

11 

6.062 
6.225 

I 

o  .019 

0.022 

37 
38 

O.IOI 

o  .104 

9 

i  28.466 

9 

•474 

39 

6.389 

9 

o  .025 

39 

o  .106 

10 

i  38.296 

10 

.638 

40 

6-553 

10 

O.O27 

40 

o  .109 

II 

i  48.125 

ii 

.802 

41 

6.717 

ii 

0.030 

41 

O.II2 

12 

13 

i  57  -955 
2     7.784 

12 

13 

.966 
2.130 

42 
43 

6.881 
7-045 

12 
13 

0.033 

°'°35 

42 
43 

O.II7 

14 

2    17  .614 

14 

2.294 

44 

7.208 

14 

0.038 

44 

0  .120 

15 

2    27.443 

IS 

2-457 

45 

7-372 

15 

0.041 

45 

0.123 

16 

2   37.273 

16 

2  .621 

46 

7.536 

16 

0.044 

46 

o  .126 

11 

2   47.102 
2    56.932 

17 
18 

2  .785 
2.949 

7.700 
7.864 

11 

0.046 
0.049 

47 
48 

0.128 
O.I3I 

19 

3     6  .762 

19 

3  -"3 

49 

8.027 

19 

0.052 

49 

0.134 

20 

3  16.591 

20 

3-277 

50 

8.191 

20 

0-055 

50 

0-137 

21 

3  26.421 

21 

3-440 

51 

8-355 

21 

0.057 

51 

0.139 

22 

3  36.250 

22 

3.604 

52 

8.519 

22 

0.060 

52 

o  .142 

23 

3  46.080 

23 

3-768 

53 

8.683 

23 

0.063 

53 

0.145 

24 

3  55  -909 

24 

3-932 

54 

8.847 

24 

o  .066 

54 

0.147 

25 

4.096 

55 

9  .010 

25 

0.068 

55 

0.150 

26 

4-259 

56 

9.174 

26 

0.071 

56 

0.153 

27 
28 

4-423 
4-587 

57 
58 

9.338 
9.502 

27 
28 

0.074 
o  .076 

11 

0.156 
0.158 

29 
30 

4-751 
4-915 

is 

9.666 
9.830 

29 
30 

0.079 
0.082 

59 
60 

o  .161 
o  .164 

342 


TABLES 


TABLE   IX.  — FOR  CONVERTING  MEAN   SOLAR  INTO 
SIDEREAL  TIME 

[Increase  in  Sun's  Right  Ascension  in  Solar  h.  m.  s.] 

Sidereal  Time  =  Mean  Time  +  C. 


Mean 
Hrs. 

Corr. 

Mean 
Min. 

Corr. 

Mean 
Min. 

Corr. 

Mean 
Sec. 

Corr. 

Mean 
Sec. 

Corr. 

I 

m        s 
o     9  .856 

I 

s 
o  .164 

31 

s 
5-°93 

I 

s 
0.003 

31 

0.085 

2 

o  19.713 

2 

0.329 

32 

5-257 

2 

o  .005 

32 

0.088 

3 

o  29.569 

3 

0-493 

33 

5-42i 

3 

0.008 

33 

o  .090 

4 

o  39  .426 

4 

0.657 

34 

5.585 

4 

0  .Oil 

34 

0.093 

5 

o  49  .282 

5 

0.821 

35 

5-750 

5 

o  .014 

35 

o  .096 

6 

o  59  -139 

6 

0.986 

36 

5.9I4 

6 

o  .016 

36 

0.099 

7 

8-995 

7 

i  .150 

37 

6.078 

7 

o  .019 

37 

0  .101 

8 

18.852 

8 

i  -3!4 

38 

6  .242 

8 

0  .022 

38 

o  .104 

9 

28.708 

9 

1.478 

39 

6.407 

9 

0.025 

39 

o  .107 

10 

38  -565 

10 

1-643 

40 

6-571 

10 

o  .027 

40 

O.IIO 

ii 

48.421 

ii 

i  .807 

4i 

6-735 

ii 

0.030 

4i 

O  .112 

12 

58-278 

12 

1.971 

42 

6  .900 

12 

0.033 

42 

O.II5 

13 

2       8.134 

13 

2.136 

43 

7.064 

13 

0.036 

43 

o  .118 

14 

2     17.991 

14 

2.300 

44 

7.228 

14 

0.038 

44 

o  .120 

15 

2    27.847 

15 

2.464 

45 

7-392 

i5 

o  .041 

45 

0.123 

16 

2    37.704 

16 

2.628 

46 

7-557 

16 

0.044 

46 

o  .126 

17 

2    47.560 

i7 

2-793 

47 

7.721 

i7 

0.047 

47 

0.129 

18 

2    57-417 

18 

2-957 

48 

7.885 

18 

0.049 

48 

0.131 

iQ 

3     7-273 

19 

3.121 

49 

8.049 

19 

0.052 

49 

0.134 

20 

3  17-129 

20 

3-285 

50 

8.214 

20 

0.055 

50 

0.137 

21 

3  26.986 

21 

3-45° 

5i 

8.378 

21 

0.057 

51 

o  .140 

22 

3  36-842 

22 

3.614 

52 

8.542 

22 

0.060 

52 

o  .142 

23 

3  46.699 

23 

3.778 

53 

8.707 

23 

0.063 

53 

0.145 

24 

3  56-555 

24 

3-943 

54 

8.871 

24 

0.066 

54 

0.148 

25 

4.107 

55 

9-035 

25 

0.068 

55 

0.151 

26 

4.271 

56 

9.199 

26 

o  .071 

56 

0.153 

27 

4-435 

57 

9-364 

27 

0.074 

57 

0.156 

28 

4  .600 

58 

9-528 

28 

0.077 

58 

0.160 

29 

4.764 

59 

9.692 

29 

0.079 

59 

0.162 

30 

4.928 

60 

9-856 

30 

o  .082 

60 

o  .16^1 

TABLES 


343 


TABLE  X.  — LENGTHS  OF   ARCS  OF  THE   PARALLEL  AND 
THE  MERIDIAN  AND   LOGS  OF  N   AND  R*» 

[Metric  Units.] 


Latitude. 

Parallel. 
Value  of  1°. 

Meridian. 
Value  of  1°. 

LogN. 

LogRm. 

0 

Meters. 

Meters. 

0    00 

111,321 

110,567.2 

6.8046985 

6.8017489 

30 

1,361 

567.3 

6987 

7493 

1    00 

1,304 

567.6 

6990 

7502 

30 

1,283 

568.0 

6996 

7519 

2    00 

1,253 

568.6 

7003 

7543 

30 

1,215 

569.4 

7012 

7573 

3    00 

1,169 

570.3 

7025 

7610 

30 

1,114 

571.4 

7040 

7654 

4    00 

1,051 

572.7 

7057 

7704 

30 

110,980 

574.1 

7076 

7761 

5    00 

110,900 

110,575.8 

6.8047097 

6.8017824 

30 

0,812 

577.6 

7120 

7894 

6    00 

0,715 

579.5 

7146 

7971 

30 

0,610 

581.6 

7174 

8054 

7    00 

0,497 

583.9 

7203 

8144 

30 

0,375 

586.4 

7235 

8240 

8    00 

0,245 

589.0 

7270 

8343 

30 

0,106 

591.8 

7307 

8452 

9    00 

109,959 

594.7 

7345 

8568 

30 

9,804 

597.8 

7385 

8690 

ID    00 

109,641 

110,601.1 

6.8047428 

6.8018819 

30 

9,469 

604.5 

7474 

8954 

11    00 

9,289 

608.1 

7520 

9094 

30 

9,101 

611.9 

7570 

9241 

12    00 

108,904 

615.8 

7620 

9395 

30 

8,699 

619.8 

7673 

9555 

13    00 

8,486 

624.1 

7729 

9720 

30 

8,265 

628.4 

7786 

9892 

14    00 

8,036 

633.0 

7845 

6.8020070 

30 

107,798 

637.6 

7907 

0254 

15    00 

107,553 

110,642.5 

6.8047970 

6.8020443 

30 

7,299 

647.5 

8035 

0639 

16    00 

7,036 

652.6 

8102 

0839 

30 

6,766 

657.8 

8171 

1047 

17    00 

6,487 

663.3 

8242 

1258 

30 

6,201 

668.8 

8315 

1477 

18    00 

5,906 

674.5 

8389 

1701 

30 

5,604 

680.4 

8465 

1930 

19    00 

5,294 

686.3 

8544 

2165 

30 

4,975 

692.4 

8624 

2404 

20    00 

104,649 

110,698.7 

6.8048705 

6.8022649 

30 

4,314 

705.1 

8789 

2900 

21    00 

3,972 

711.6 

8874 

3155 

30 

3,622 

718.2 

8960 

3415 

22    00 

3,264 

725.0 

9049 

3680 

.30 

2,898 

731.8 

9139 

3950 

344 


TABLES 


TABLE  X   (Con.)  —  LENGTHS  OF  ARCS  OF  THE   PARALLEL 
AND  THE  MERIDIAN   AND   LOGS   OF   N   AND   Rm 

IMetric  Units.] 


Latitude. 

Parallel. 
Value  of  1°. 

Meridian. 
Value  of  1°. 

LogN. 

Log  Rm. 

0 

Meters. 

Meters. 

23    00 

102,524 

110,738.8 

6.8029231 

6.8044225 

30 

2,143 

746.0 

9323 

4504 

24    00 

1,754 

753.2 

9418 

4788 

30 

1,357 

760.6 

9514 

5077 

25    00 

100,952 

110,768.0 

6.8049612 

6.8025370 

30 

0,539 

775.6 

9711 

5667 

26    00 

0,119 

783.3 

9812 

5968 

30 

99,692 

791.1 

9914 

6274 

27    00 

9,257 

799.0 

6.8050017 

6584 

30 

8,814 

807.0 

0121 

6897 

28    00 

8,364 

815.1 

0227 

7215 

30 

7,906 

823.3 

0334 

7536 

29    00 

7,441 

831.6 

0443 

7862 

30 

6,968 

840.0 

0552 

8190 

30    00 

96,488 

110,848.5 

6.8050663 

6.8028522 

30 

6,001 

857.0 

0774 

8857 

31    00 

95,506 

865.7 

0888 

9197 

30 

5,004 

874.4 

1002 

9539 

32    00 

4,495 

883.2 

1117 

9883 

30 

3,979 

892.1 

1233 

6.8030231 

33    00 

3,455 

901.1 

1350 

0582 

30 

2,925 

910.1 

1468 

0935 

34    00 

2,387 

919.2 

1586 

1292 

30 

1,842 

928.3 

1706 

1651 

35    00 

91,290 

110,937.6 

6.8051826 

6.8032012 

30 

0,731 

946.9 

1947 

2375 

36    00 

0,166 

956.2 

2069 

2741 

30 

89.593 

965.6 

2192 

3109 

37    00 

9i014 

975.1 

2315 

3479 

30 

8,428 

984.5 

2439 

3850, 

38    00 

7,835 

994.1 

2564. 

4224 

30 

7,235 

111,003.7 

2689 

4599 

39    00 

6,629 

013.3 

2814 

4976 

30 

6,016 

023.0 

2940 

5354 

40    00 

85,396 

111,032.7 

6.8053067  / 

6.8035734  < 

30 

4,770 

042.4 

3194 

6115  < 

41    00 

4,137 

052.2 

3321 

6496 

30 

3,498 

061.9 

3448 

6878 

42    00 

2,853 

071.7 

3576 

7262 

30 

2,201 

081.6 

3704 

7646 

43    00 

1,543 

091.4 

3832 

8031 

30 

0,879 

101.3 

3960 

8416 

.    44    00 

80,208 

111.1 

4089 

8802 

30 

79,532 

121.0 

4218 

9188 

45    00 

78,849 

111,130.9 

6.8054347 

6.8039574 

30 

8,160 

140.8 

4476 

9960 

TABLES 


345 


TABLE  X   (Con.)-  — LENGTHS  OF  ARCS  OF  THE  PARALLEL 
AND  THE  MERIDIAN   AND  LOGS  OF  N  AND  R« 

[Metric  Units.] 


Latitude. 

Parallel. 
Value  of  1°. 

Meridian. 
Value  of  1°. 

LogN. 

LogRm. 

1                  0 

Meters. 

Meters. 

46    00 

77,466 

111,150.6 

6.8054604 

6.8040346 

30 

6,765 

160.5 

4732 

0731 

47    00 

6,058 

170.4 

4861 

1117 

30 

5,346 

180.2 

4989 

1502 

48    00 

4,628 

190.1 

5118 

1887 

30 

3,904 

199.9 

5246 

2270 

49    00 

3,174 

209.7 

5373 

2653 

30 

2,439 

219.5 

5500 

3034 

50    00 

71,698 

111,229.3 

6.8055628 

6.  80434*16 

30 

0,952 

239.0 

5754 

3796 

51    00 

0,200 

248.7 

5880 

4175 

•30 

69,443 

258.3 

6006 

4552 

52    00 

8,680 

268.0 

6131 

4928 

30 

7,913 

277.6 

6256 

5302 

53    00 

7,140 

287.1 

6380 

5674 

30 

6,361 

296.6 

6504 

6044 

54    00 

5,578 

306.0 

6627 

6413 

30 

4,790 

315.4 

6749 

6779 

55    00 

63,996 

111,324.8 

6.8056870 

6.8047144 

30 

3,198 

334.0 

6991 

7506 

56    00 

2,395 

343.3 

7111 

7866 

30 

1,587 

352.4 

7230 

8223 

57    00 

0,774 

361.5 

7348 

8578 

30 

59,957 

370.5 

7465 

8929 

58    00 

9,135 

379.5 

7582 

9279 

30 

8,309 

388.4 

7697 

9624 

59    00 

7,478 

397.2 

7811 

9968 

30 

6,642 

405.9 

7924 

6.8050307 

60    00 

55,802 

111,414.5 

6.8058037 

6.8050644 

30 

4,958 

423.1 

8148 

0977 

61    00 

4,110 

431.5 

8258 

1307 

30 

.   3,257 

439.9 

8366 

1633 

62    00 

2,400 

448.2 

8474 

1956 

30 

1,540 

456.4 

8580 

2274 

63    00 

0,675 

464.4 

8685 

2590 

30 

49,806 

472.4 

8789 

2900 

64    00 

8,934 

480.3 

8891 

3208 

30 

8,057 

488.1 

8992 

3510 

65    00 

47,177 

111,495.7 

6.8059092 

6.8053809 

30 

6,294 

503.3 

9190 

4103 

66    00 

5,407 

510.7 

9287 

4393 

30 

4,516 

518.0 

9382 

4678 

67    00 

43,622 

525.3 

9475 

4959 

30 

2,724 

532.3 

9567 

5235 

68    00 

1,823 

539.3 

9658 

5506* 

30 

0,919 

546.2 

9747 

5772 

346 


TABLES 


TABLE  X   (Cow.).  — LENGTHS  OF  ARCS  OF  THE   PARALLEL 
AND  THE  MERIDIAN   AND   LOGS   OF   N   AND   Rm 

[Metric  Units.] 


Latitude. 

Parallel. 
Value  of  1°. 

Meridian. 
Value  of  1°.. 

LogN. 

Log  Rm. 

, 

Meters. 

Meters. 

69    00 

40,012 

111,552.9 

6.8069834 

6.8056034 

30 

39,102 

559.5 

9919 

6290 

70    00 

38,188 

111,565.9 

6.8060003 

6.8056542 

30 

7,272 

572.2 

0085 

6788 

71    00 

6,353 

578.4 

0165 

7029 

30 

5,421 

584.5 

0244 

7264 

72    00 

4,506 

590.4 

0321 

7495 

30 

3,578 

596.2 

0396 

7719 

73    00 

2,648 

601.8 

0468 

7938 

30 

1,716 

607.3 

0539 

8153 

74    00. 

0,781 

612.7 

0608 

8361 

30 

29,843 

617.9 

0676 

8563 

75    00 

28,903 

111,622.9 

6.8060742 

6.8058759 

30 

7,961 

627.8 

0805 

8950 

76    00 

7,017 

632.6    • 

0867 

9135 

30 

6,071 

637.1 

0927 

9314 

77    00 

5,123 

641.6 

0984 

9487 

30 

4,172 

645.9 

1040 

9653 

78    00 

3,220 

650.0 

1093 

9814 

30 

2,266 

653.9 

1145 

9968 

79    00 

1,311 

657.8 

1195 

6.8060118 

30 

20,353 

661.4 

1242 

0258 

80    00 

19,394 

111,664.9 

6.8061287 

6.8060394 

30 

8,434 

668.2 

1330 

0523 

81    00 

7,472 

671.4 

1371 

'0646 

30 

6,509 

674.4 

1409 

0763 

82    00 

5,545 

677.2 

1446 

0873 

30 

4,579 

679.9 

1480 

0976 

83    00 

3,612 

682.4 

1513 

1074 

30 

2,644 

684.7 

1544 

1163 

84    00 

1,675 

686.9 

1571 

1248 

30 

10,706 

688.9 

1597 

1325 

85    00 

9,735 

111,690.7 

6.8061620 

6.8061395 

30 

8,764 

692.3 

1642 

1459 

86    00 

7,792 

693.8 

1661 

1517 

30 

6,819 

695.1 

1678 

1567 

87    00 

5,846 

696.2 

1692 

1611 

30 

4,872 

697.2 

1705 

1648 

88    00 

3,898 

697.9 

1715 

1679 

30 

2,924 

698.6 

1723 

1702 

89    00 

1,949 

699.0 

1728 

1719 

30 

975 

699.3 

1731 

1729 

90    00 

0 

111,699.3 

6.8061733 

6.8061733 

TABLES 


347 


TABLE  XL —  TABLE  OF  LOGARITHMS  OF  RADII  OF  CURVA- 
TURE OF  THE  EARTH'S  SURFACE  IN  METERS  FOR  VARIOUS 
LATITUDES  AND  AZIMUTHS 

[Based  upon  Clarke's  Ellipsoid  of  Rotation  (1866)  .J 


Azimuth. 

0°  lat. 

l°lat. 

2°  lat. 

3°  lat. 

4°  lat. 

5°  lat. 

6°  lat. 

0 

Meridian. 

6.80175 

6.80175 

6.80175 

6.80176 

6.80177 

6.80178 

6.80180 

5 

177 

177 

178 

178 

179 

180 

182 

10 

184 

184 

184 

185 

186 

187 

188 

15 

195 

195 

195 

196 

197 

198 

199 

20 

209 

209 

210 

210 

211 

212 

214 

25 

227 

228 

228 

228 

229 

230 

232 

30 

248 

249 

249 

250 

250 

251 

252 

35 

272 

272 

272 

273 

273 

274 

276 

40 

296 

297 

297 

297 

298 

299 

300 

45 

322 

322 

322 

323 

324 

324 

325 

50 

348 

348 

348 

348 

349 

350 

351 

55 

373 

373 

373 

373 

374 

374 

375 

60 

396 

396 

396 

396 

397 

398 

398 

65 

417 

417 

417 

418 

418 

418 

419 

70 

435 

435 

436 

436 

436 

437 

437 

75 

450 

450 

450 

450 

451 

451 

452 

80 

461 

461 

461 

461 

462 

462 

463 

85 

468 

468 

468 

468 

468 

469 

469 

90 

470 

470 

470 

470 

471 

471 

472 

Azimuth. 

6°  lat. 

7°  lat. 

8°  lat. 

9°  lat. 

10°  lat. 

11°  lat. 

12°  lat. 

Meridian. 

6.80180 

6.80181 

6.80183 

6.80186 

6.80188 

6.80191 

6.80194 

5 

182 

184 

186 

188 

190 

193 

196 

10 

188 

190 

192 

194 

197 

200 

202 

15 

199 

201 

203 

205 

207 

210 

213 

20 

214 

215 

217 

219 

222 

224 

227 

25 

232 

233 

235 

237 

239 

242 

244 

30 

252 

254 

256 

257 

260 

262 

264 

35 

276 

277 

278 

280 

282 

284 

287 

40 

300 

301 

303 

304 

306 

308 

310 

45 

325 

326 

328 

329 

331 

333 

335 

50 

351 

352 

353 

354 

356 

358 

359 

55 

375 

376 

377 

379 

380 

382 

383 

60 

398 

399 

400 

401 

403 

404 

406 

65 

419 

420 

421 

422 

423 

424 

426 

70 

437 

438 

439 

440 

441 

442 

443 

75 

452 

452 

453 

454 

455 

456 

457 

80 

463 

463 

464 

465 

466 

467 

468 

85 

469 

470 

470 

471 

472 

473 

474 

90 

472 

472 

473 

474 

474 

475 

476 

348 


TABLES 


TABLE  XI  (Cow.).  — TABLE  OF  LOGARITHMS  OF  RADII  OF 
CURVATURE  OF  THE  EARTH'S  SURFACE  IN  METERS  FOR 
VARIOUS  LATITUDES  AND  AZIMUTHS 

[Based  upon  Clarke's  Ellipsoid  of  Rotation  (1866).] 


Azimuth. 

12°  lat. 

13°  lat. 

14°  lat. 

15°  lat. 

16°  lat. 

17°  lat. 

18°  lat. 

o 

Meridian. 

6.80194 

6.80197 

6.80201 

6.80204 

6.80208 

6.80213 

6.80217 

5 

196 

199 

203 

206 

210 

215 

219 

10 

202 

206 

209 

213 

217 

221 

225 

15 

213 

216 

219 

223 

227 

231 

235 

20 

227 

230 

233 

236 

240 

244 

248 

25 

244 

247 

250 

254 

257 

261 

265 

30 

264 

267 

270 

273 

276 

280 

284 

35 

287 

289 

292 

295 

298 

301 

305 

40 

310 

313 

315 

318 

321 

324 

327 

45 

335 

337 

339 

342 

344 

347 

350 

50 

359 

361 

364 

366 

368 

371 

373 

55 

383 

385 

387 

389 

391 

394 

396 

60 

406 

407 

409 

411 

413 

415 

417 

65 

426 

427 

429 

430 

432 

434 

436 

70 

443 

444 

446 

447 

449 

451 

453 

75 

457 

458 

460 

461 

463 

464 

466 

80 

468 

469 

470 

471 

473 

474 

476 

85 

474 

475 

476 

478 

479 

480 

482 

90 

476 

'  477 

478 

480 

481 

482 

484 

•  Azimuth. 

18°  lat. 

19°  lat. 

20°  lat. 

21°  lat. 

22°  lat. 

23  Mat. 

24°  lat. 

6 

Meridian. 

6.80217 

6.80222 

6.80226 

6.80232 

6.80237 

6.80242 

6.80248 

5 

219 

224 

228 

234 

239 

244 

250 

10 

225 

230 

234 

239 

244 

250 

255 

15 

235 

239 

244 

249 

254 

259 

264 

20 

248 

252 

257 

262 

266 

271 

277 

25 

265 

269 

273 

277 

282 

287 

292 

30 

284 

287 

292 

296 

300 

305 

309 

35 

305 

308 

312 

316 

320 

324 

329 

40 

327 

330 

334 

338 

341 

345 

350 

45 

350 

353 

357 

360 

364 

367 

371 

50 

373 

376 

379 

382 

386 

389 

392 

55 

396 

398 

401 

404 

407 

410 

413 

60 

417 

419 

422 

424 

427 

430 

432 

65 

436 

438 

440 

443 

445 

448 

450 

70 

453 

454 

456 

459 

461 

463 

465 

75 

466 

468 

470 

472 

473 

476 

478 

80 

476 

478 

479 

481 

483 

485 

487 

85 

482 

483 

485 

487 

489 

490 

492 

90 

484 

485 

487 

489 

490 

492 

494 

TABLES 


349 


TABLE  XI  (Con.).  — TABLE  OF  LOGARITHMS  OF  RADII  OF 
CURVATURE  OF  THE  EARTH'S  SURFACE  IN  METERS  FOR 
VARIOUS  LATITUDES  AND  AZIMUTHS 

[Based  upon  Clarke's  Ellipsoid  of  Rotation  (1866).] 


Azimuth. 

24°  lat. 

25°  lat. 

26°  lat. 

27°  lat. 

28°  lat. 

29°  lat. 

30°  lat. 

Meridian. 

6.80248 

6.80254 

6.80260 

6.80266 

6.80272 

6.80279 

6.80285 

5 

250 

256 

262 

268 

274 

280 

287 

10 

255 

261 

267 

273 

279 

285 

292 

15 

264 

270 

276 

282 

288 

294 

300 

20 

277 

282 

288 

293 

299 

305 

311 

25 

292 

297 

302 

308 

313 

319 

325 

30 

309 

314 

319 

324 

330 

335 

340 

35 

329 

333 

338 

343 

348 

353 

358 

40 

350 

354 

358 

362 

367 

372 

377 

45 

371 

375 

379 

383 

387 

391 

396 

50 

392 

396 

399 

403 

407 

411 

415 

55 

413 

416 

420 

423 

426 

430 

434 

60 

432 

435 

438 

442 

445 

448 

451 

65 

450 

453 

455 

458 

461 

464 

467 

70 

465 

468 

470 

473 

475 

478 

481 

75 

478 

480 

482 

484 

487 

489 

492 

80 

487 

489 

491 

493 

495 

498 

500 

85 

492 

494 

496 

498 

501 

503 

505 

90 

494 

496 

498 

500 

502 

504 

507 

Azimuth. 

30°  lat. 

31°  lat. 

32°  lat. 

33°  lat. 

34°  lat. 

35°  lat. 

36°  lat. 

Meridian. 

6.80285 

6.80292 

6.80299 

6.80306 

6.80313 

6.80320 

6.80327 

5 

287 

294 

300 

307 

314 

322 

329 

10 

292 

298 

305 

312 

319 

326 

333 

15 

300 

306 

313 

320 

326 

333 

340 

20 

311 

317 

324 

330 

337 

343 

350 

25 

325 

331 

337 

343 

349 

355 

362 

30 

340 

346 

352 

358 

364 

370 

376 

35 

358 

363 

369 

374 

380 

385 

391 

40 

377 

382 

386 

392 

397 

402 

407 

45 

396 

400 

405 

410 

414 

419 

424 

50 

415 

419 

423 

428 

432 

436 

441 

55 

434 

437 

441 

445 

449 

453 

457 

60 

451 

455 

458 

462 

465 

469 

472 

65 

467 

470 

473 

476 

480 

483 

486 

70 

481 

484 

486 

489 

492 

495 

498 

75 

492 

494 

497 

500 

502 

505 

508 

80 

500 

502 

505 

507 

510 

512 

515 

85 

505 

507 

510 

512 

514 

517 

519 

90 

507 

509 

511 

514 

516 

518 

521 

350 


TABLES 


TABLE  XI  (Cow.).  — TABLE  OF  LOGARITHMS  OF  RADII  OF 
CURVATURE  OF  THE  EARTH'S  SURFACE  IN  METERS  FOR 
VARIOUS  LATITUDES  AND  AZIMUTHS 

[Based  upon  Clarke's  Ellipsoid  of  Rotation  (1866).] 


Azimuth. 

36°  lat. 

37°  lat. 

38°  lat. 

39°  lat. 

40°  lat. 

41°  lat. 

42°  lat. 

Meridian. 

6.80327 

6.80335 

6.80342 

6.80350 

6.80357 

6.80365 

6.80373 

5 

329 

336 

344 

351 

359 

366 

374 

10 

333 

340 

348 

355 

363 

370 

378 

15 

340 

348 

355 

362 

369 

376 

384 

20 

350 

357 

364 

371 

378 

385 

392 

25 

362 

368 

375 

382 

388 

395 

402 

30 

376 

382 

388 

394 

401 

407 

413 

35 

391 

397 

402 

408 

414 

420 

426 

40 

407 

412 

418 

423 

429 

434 

440 

45 

424 

429 

434 

439 

444 

449 

454 

50 

441 

445 

450 

454 

459 

464 

468 

55 

457 

461 

465 

469 

474 

478 

482 

60 

472 

476 

480 

484 

487 

491 

495 

65 

486 

489 

493 

496 

500 

503 

507 

70 

498 

501 

504 

507 

510 

514 

517 

75 

508 

510 

513 

516 

519 

522 

525 

80 

515 

517 

520 

523 

525 

528 

531 

85 

519 

522 

524 

527 

529 

532 

534 

90 

521 

523 

526 

528 

531 

533 

536 

Azimuth. 

42°  lat. 

43°  lat. 

44°  lat. 

45°  lat. 

46°  lat. 

47°  lat. 

48°  lat. 

Meridian. 

6.80373 

6.80380 

6.80388 

6.80396 

6.80404 

6.80411 

6.80419 

5 

374 

382 

389 

397 

404 

412 

420 

10 

378 

385 

393 

400 

408 

415 

423 

15 

384 

391 

398 

406 

413 

420 

428 

20 

392 

399 

406 

413 

420 

427 

434 

25 

402 

408 

415 

422 

429 

436 

442 

30 

413 

420 

426 

433 

439 

446 

452 

35 

426 

432 

438 

444 

450 

456 

462 

40 

440 

446 

451 

457 

462 

468 

474 

45 

454 

459 

464 

470 

475 

480 

485 

50 

468 

473 

478 

482 

487 

492 

496 

55 

482 

486 

490 

495 

499 

503 

508 

60 

495 

499 

502 

506 

510 

514 

518 

65 

507 

510 

514 

517 

520 

524 

528 

70 

517 

520 

523 

526 

529 

532 

536 

75 

525 

528 

531 

534 

536 

539 

542 

80 

531 

534 

536 

539 

542 

544 

547 

85 

534 

537 

540 

542 

545 

548 

550 

90 

536 

538 

541 

544 

546 

549 

551 

TABLES 


351 


TABLE  XI  (Con.)-  — TABLE  OF  LOGARITHMS  OF  RADII  OF 
CURVATURE  OF  THE  EARTH'S  SURFACE  IN  METERS  FOR 
VARIOUS  LATITUDES  AND  AZIMUTHS 

[Based  upon  Clarke's  Ellipsoid  of  Rotation  (1866).] 


Azimuth. 

48°  lat. 

49°  lat. 

50°  lat. 

51°  lat. 

52°  lat. 

53°  lat. 

54°  lat. 

Meridian. 

6.80419 

6.80426 

6.80434 

6.80442 

6.80449 

6.80457 

6.80464 

5 

420 

428 

435 

443 

450 

458 

465 

10 

423 

430 

438 

445 

453 

460 

467 

15 

428 

435 

442 

450 

457 

464 

471 

20 

434 

441 

448 

455 

462 

469 

476 

25 

442 

449 

456 

463 

469 

476 

482 

30 

452 

458 

465 

471 

477 

484 

490 

35 

462 

468 

474 

480 

486 

492 

498 

40 

474 

479 

485 

490 

496 

.  501 

506 

45 

485 

490 

495 

500 

505 

510 

515 

50 

496 

501 

506 

510 

515 

520 

524 

55 

508 

512 

516 

520 

524 

528 

533 

60 

518 

522 

526 

530 

533 

537 

541 

65 

528 

531 

534 

538 

541 

545 

548 

70 

536 

539 

542 

545 

548 

551 

554 

75 

542 

545 

548 

551 

554 

557 

559 

80 

547 

550 

553 

555 

558 

561 

563 

85 

550 

553 

555 

558 

560 

563 

566 

90 

551 

554 

556 

559 

561 

564 

566 

Azimuth. 

54°  lat. 

55°  lat. 

56°  lat. 

57°  lat. 

58°  lat. 

59°  lat. 

60°  lat. 

Meridian. 

6.80464 

6.80471 

6.80479 

6.80486 

6.80493 

6.80500 

6.80506 

5 

465 

472 

479 

486 

493 

500 

07 

10 

467 

474 

481 

488 

495 

502 

09 

15 

471 

478 

485 

492 

498 

505 

11 

20 

476 

483 

489 

496 

502 

509 

15 

25 

482 

489 

495 

501 

508 

514 

20 

30 

490 

496 

502 

508 

514 

519 

25 

35 

498 

503 

509 

515 

520 

525 

31 

,  40 

506 

512 

517 

522 

527 

532 

37 

45 

515 

520 

525 

530 

534 

539 

43 

50 

524 

528 

533 

537 

542 

546 

50 

55 

533 

537 

541 

545 

548 

552 

56 

60 

541 

544 

548 

552 

555 

558 

62 

65 

548 

551 

555 

558 

561 

564 

67 

70 

554 

557 

560 

563 

566 

569 

72 

75 

559 

562 

565 

568 

570 

573 

75 

80 

563 

566 

568 

571 

573 

576 

78 

85 

566 

568 

570 

573 

575 

578 

80 

90 

566 

569 

571 

574 

576 

578 

80 

352 


TABLES 


TABLE  XI  (Cow.)-  — TABLE  OF  LOGARITHMS  OF  RADII  OF 
CURVATURE  OF  THE  EARTH'S  SURFACE  IN  METERS  FOR 
VARIOUS  LATITUDES  AND  AZIMUTHS 

[Based  upon  Clarke's  Ellipsoid  of  Rotation  (1866).] 


Azimuth. 

60°  lat. 

61°  lat. 

62°  lat. 

63°  lat. 

64°  lat. 

65°  lat. 

66°  lat. 

0 

Meridian. 

6.80506 

6.80513 

6.80520 

6.80526 

6.80532 

6.80538 

6.80544 

5 

07 

14 

20 

26 

32 

38 

44 

10 

09 

15 

22 

28 

34 

40 

45 

15 

11 

18 

24 

30 

36 

42 

47 

20 

15 

21 

27 

33 

39 

44 

50 

25 

20 

26 

31 

37 

42 

48 

53 

30 

25 

30 

36 

41 

46 

51 

56 

35 

31 

36 

41 

46 

51 

56 

60 

40 

•  37 

42 

46 

51 

56 

60 

64 

45 

43 

48 

52 

56 

60 

64 

68 

50 

50 

54 

58 

62 

65 

69 

73 

55 

56 

60 

63 

67 

70 

74 

77 

60 

62 

65 

68 

72 

75 

78 

81 

65 

67 

70 

73 

76 

79 

82 

84 

70 

72 

74 

77 

80 

82 

85 

87 

75 

75 

78 

80 

83 

85 

87 

90 

80 

78 

80 

83 

85 

87 

89 

91 

85 

80 

82 

84 

86 

88 

90 

92 

90 

80 

83 

85 

87 

89 

91 

93 

Azimuth. 

66°  lat. 

67°  lat. 

68°  lat. 

69°  lat. 

70°  lat. 

71°  lat. 

72°  lat. 

Meridian. 

6.80544 

6.80550 

6.80555 

6.80560 

6.80565 

6.80570 

6.80575 

5 

44 

50 

55 

61 

66 

70 

75 

10 

45 

51 

56 

62 

66 

71 

76 

15 

47 

53 

58 

63 

68 

72 

77 

20 

50 

55 

60 

65 

70 

74 

78 

25 

53 

58 

62 

67 

72 

76 

80 

30 

56 

61 

65 

70 

74 

•     78 

82 

35 

60 

64 

69 

73 

77 

81 

84 

40 

64 

68 

72 

76 

80 

83 

87 

45 

68 

72 

76 

79 

83 

86 

89 

50 

73 

76 

79 

83 

86 

89 

92 

55 

77 

80 

83 

86 

89 

91 

94 

60 

81 

84 

86 

89 

91 

94 

96 

65 

84 

87 

89 

92 

94 

96 

98 

70 

87 

90 

92 

94 

96 

98 

6.80600 

75 

90 

92 

94 

96 

98 

6.80600 

01 

80 

91 

93 

95 

97 

99 

01 

02 

85 

92 

94 

96 

98 

6.80600 

01 

03 

90 

93 

95 

97 

98 

00 

02 

03 

TABLES 


353 


TABLE  XII.  — VALUES   OF   LOG  m  FOR  COMPUTING   SPHERI- 
CAL EXCESS.    (METRIC  SYSTEM.) 


Latitude 

Log  m 

Latitude 

Log  m 

Latitude 

Log  m 

o   / 
18  oo 

I  .40639-  10 

0    / 

33  °° 

i  .40520  —  10 

0    / 

48  oo 

i  .40369  -  10 

18  30 

636 

33  3° 

516 

48  30 

364 

19  oo 

632 

34  oo 

5" 

49  oo 

359 

19  30 

629 

34  3° 

506 

49  3° 

354 

2O  00 

626 

35  o° 

5oi 

50  oo 

349 

20  30 

623 

35  30 

496 

5°  3° 

344 

21  OO 

619 

36  oo 

491 

51  oo 

339 

21  30 

616 

36  3° 

486 

5i  3° 

334 

22  00 

612 

37  oo 

482 

52  oo 

329 

22  30 

608 

37  3° 

477 

52  3° 

324 

23  oo 

605 

38  oo 

472 

53  oo 

319 

23  3° 

601 

38  30 

467 

53  3° 

3H 

24  oo 

597 

39  °° 

462 

54  oo 

309 

24  30 

594 

39  3° 

457 

54  3° 

304 

25  oo 

590 

40  oo 

452 

55  oo 

299 

25  3° 

586 

40  30 

446 

55  30 

295 

26  oo 

582 

41  oo 

441 

56  oo 

290 

26  30 

578 

4i  3° 

436 

56  30 

285 

27  oo 

573 

42  oo 

43  1 

57  oo 

280 

27  3° 

569 

42  3° 

426 

57  30 

276 

28  oo 

565 

43  °° 

421 

58  oo 

271 

28  30 

560 

43  3° 

416 

5830 

266 

29  oo 

556 

44  oo 

411 

59  oo 

262 

29  30 

SS2 

44  3° 

406 

59  3° 

*57 

30  oo 

548 

45  °° 

400 

60  oo 

253 

30  3° 

544 

45  3° 

395 

60  30 

249 

31  oo 

539 

46  oo 

390 

61  oo 

244 

31  30 

534 

46  30 

385 

61  30 

240 

32  oo 

53° 

47  oo 

380 

62  oo 

235 

32  3° 

i  .40525 

47  3° 

i  -40375 

62  30 

i  .40231 

(The  above  table  is  computed  for  the  Clarke  spheroid  of  1866.) 


354 


TABLES 


TABLE    XIII.  — CORRECTION    TO    LONGITUDE    FOR    DIFFER- 
ENCE BETWEEN  ARC  AND   SINE 


logs(-). 

log  difference. 

logdX(+) 

logs(-). 

log  difference, 

log  d\  (+). 

3  -876 

o  .000  oooi 

2-385 

4-871 

o  .000  0098 

3-38o 

4  .026 

02 

2-535 

4.882 

103 

3-391 

4.114 

03 

2  .623 

4.892 

1  08 

3.401 

4.177 

04 

2.686 

4.903 

114 

3-412 

4.225 

°5 

2.734 

4.913 

119 

3.422 

4  -265 

06 

2-774 

4.922 

124 

3-431 

4.298 

07 

2.807 

4.932 

130 

3-441 

4.327 

08 

2.836 

4.941 

136 

3-450 

4-353 

09 

2.862 

4-95° 

142 

3-459 

4.376 

10 

2.885 

4-959 

147 

3.468 

4.396 

II 

2-905 

4.968 

153 

3-477 

4-4I5 

12 

2.924 

4-976 

160 

3-485 

4-433 

13 

2.942 

4-985 

166 

3-494 

4-449 

14 

2-958 

4-993 

172 

3-502 

4-464 

15 

2-973 

5.002 

179 

3-5ii 

4.478 

16 

2.987 

5  .010 

186 

3-5I9 

4.491 

17 

3  .000 

5-oi7 

192 

3-526 

4.503 

18 

3.012 

5-025 

199 

3-534 

4.526 

20 

3-°35 

5-033 

206 

3-542 

4.548 

23 

3.057 

5.040 

213 

3-549 

4-570 

25 

3.079 

5-047 

221 

3.556 

4.591 

27 

3  .100 

5-054 

228 

3-563 

4.612 

30 

3.121 

5  -062 

236 

3-57i 

4-631 

33 

3.140 

5.068 

243 

3-577 

4.649 

36 

3-158 

5-075 

25  1 

3-584 

4-667 

39 

3-I76 

5  .082 

259 

3-591 

4.684 

42 

3-193 

5.088 

267 

3-597 

4.701 

45 

3-210 

5-095 

275 

3.604 

4.716 

48 

3-225 

5  .102 

284 

3.611 

4-732 

52 

3.241 

5.108 

292 

3.617 

4.746 

56 

3-255 

5  -ii4 

300 

3-623 

4.761 

59 

3.270 

5  -120 

3°9 

3.629 

4-774 

63 

3.283 

5  -126 

318 

3-635 

4-788 

67 

3.297 

5-132 

327 

3-641 

4.801 

7i 

3-310 

5-138 

336 

3-647 

4-813 
4-825 

6 

3.322 

3-334 

5-144 
5-15° 

345 
354 

3-653 
3-659 

4-834 

84 

3-343 

S-iS6 

364 

3-665 

4.849 

89 

3.358 

5  .161 

373 

3.670 

4.860 

94 

3-369 

5-167 

383 

3.676 

TABLES 


355 


TABLE    XIV.  —  LOGARITHMS    OF    FACTORS    FOR    COMPUTING 
GEODETIC   POSITIONS 


Lat. 

Log  A 

Log£ 

LogC 

Log£> 

LogE 

0    1 

18  oo 

—  10 
8.509  5862 

—  10 

8.512  2550 

—  10 

0.91816 

—  IO 
2  .1606 

—  20 
5.7317 

10 

5836 

2474 

o  .92243 

2  .1641 

5  -7337 

20 

5811 

2397 

o  .92667 

2  .1675 

5  -7358 

30 

5785 

2320 

0.93088 

2.1709 

5-7379 

40 

5759 

2243 

o  .93505 

2  .1742 

5-7400 

50 

5733 

2165 

o  .93919 

2-1775 

5  -7422 

19  oo 

5707 

2086 

o  -9433° 

2.l8o8 

5-7443 

10 

5681 

2006 

0.94737 

2  .1840 

5  -7464 

20 

5654 

1927 

o  .95142 

2  .1872 

5  7486 

30 

5627 

1847 

o  .95544 

2  .1903 

5  -7508 

40 

5600 

1766 

o  -95943 

2  .1934 

5  -753° 

50 

5573 

1684 

o  .96339 

2  .1965 

57552 

20  00 

5546 

1602 

o  .96733 

2  .1996 

5-7574 

10 

55i8 

1519 

0.97123 

2  .2O26 

5  .7597 

20 

5490 

1435 

0.97511 

2  .2055 

5  -7619 

30 

5462 

I35i 

0.97896 

2  .2084 

5  -7642 

40 

5434 

1267 

o  .98279 

2.2II3 

5  -7664 

50 

5406 

1182 

o  .98659 

2  .2142 

5-7688 

21  00 

5377 

1096 

o  .99037 

2  .2I7O 

5-77" 

10 

5348 

IOIO 

0.99412 

2  .2198 

5-7734 

20 

S320 

0924 

0.99785 

2  .2226 

5-7757 

30 

5290 

0836 

1  .00156 

2  .2253 

5  .778o 

40 

5261 

0748 

I  .00524 

2  .2280 

5.7804 

50 

5232 

0660 

I  .00890 

2  .2307 

5  .7828 

22  00 

5202 

O57r 

I  .01253 

2  .2333 

5  -7851 

10 

5172 

0481 

1.01615 

2  -2359 

5  -7875 

20 

5142 

0391 

1  .01974 

2.2385 

5-7899 

30 

5112 

0301 

1  .02331 

2  .2411 

5  -7924 

40 

5082 

O2  10 

1  .02686 

2  .2436 

5  -7948 

50 

5051 

0118 

I  .03039 

2  .2461 

5  -7972 

23  oo 

5020 

8.512  0026 

1  .03390 

2  .2485 

5  -7997 

10 

4990 

8-5H9934 

1  .03739 

2  .25IO 

5  .8021 

20 

4959 

9840 

1  .04086 

2  .2534 

5.8046 

30 

4927 

9747 

I  .04431 

2-2557 

5.8071 

40 

4896 

9653 

1  .04775 

2  .2581 

5-8096 

50 

4865 

9558 

1  .05116 

2  .2004 

5.8121 

24  oo 

4833 

9463 

1  .05456 

2  .2627 

5  .8146 

10 

4801 

9367 

1  .05794 

2  .2650 

5.8172 

20 

4769 

9271 

1  .06130 

2  .2672 

5-8i97 

30 

4737 

9174 

1  .06464 

2.2694 

5  -8223 

40 

4704 

9077 

1  .06797 

2  .2716 

5  -8249 

50 

4672 

8979 

1  .07128 

2  -2738 

5  -8274 

60 

8.509  4639 

8.5118881 

I  -07457 

2  .2759 

5-8300 

356 


TABLES 


TABLE  XIV   (Continued) 


Lat. 

Log  A 

LogS 

LogC 

Log£> 

LogE 

o   / 

25  oo 

8.509  4639 

8.5118881 

•07457 

2  .2759 

5  -8300 

IO 

4606 

8783 

•07785 

2  .2780 

5  -8326 

20 

4573 

8684 

.08111 

2  .2801 

5  -8352 

30 

4540 

8584 

•08435 

2  .2822 

5  -8379 

40 

45°7   - 

8484 

.08758 

2  .2842 

5  -8405 

50 

4473 

8383 

.09080 

2  .2862 

5  -8431 

26  oo 

4439 

8283 

.09400 

2  .2882 

5-8458 

10 

4406 

8181 

.09718 

2  .2902 

5  -8485 

20 

4372 

8079 

.10036 

2  .2922 

5  -8512 

3<> 

4337 

7977 

•I035I 

2  .2941 

5  -8539 

40 

43°3 

7874 

.10666 

2  .2960 

5  -8566 

50 

4269 

7771 

.10979 

2  .2978 

5  -8593 

27  oo 

4234 

7667 

.11290 

2  .2997 

5  .8620 

10 

4200 

7563 

.11600 

2  .3015 

5  .8647 

20 

4165 

7458 

.11909 

2  .3033 

5  -8675 

30 

413° 

7353 

.12217 

2  .3051 

5  -8702 

40 

4094 

7248 

•12523 

2.3069 

5  -8730 

50 

4059 

7142 

.12829 

2  .3086 

5.8757 

28  oo 

4024 

7036 

.I3I32 

2  .3104 

5  -8785 

IO 

3988 

6929 

•13435 

2.3I2I 

5-8813 

20 

3952 

6822 

•13737 

2  -3137 

5  -8841 

30 

39i7 

6714 

.14037 

2.3154 

5  -8870 

40 

3881 

6607 

•14337 

2.3170 

5-8898 

50 

3845 

6498 

•14635 

2.3187 

5  -8926 

29  oo 

3808 

6389 

•14932 

2  .3203 

5  -8955 

IO 

3772 

6280 

.15228 

2.3218 

5  -8983 

20 

3735 

6171 

•15522 

2  .3234 

5  -9012 

30 

3699 

6061 

.15816 

2  .3249 

5  -9041 

40 

3662 

595° 

.16109 

2  .3264 

5.9069 

50 

3625 

5840 

.16401 

2  .3279 

5.9098 

30  oo 

3588 

5729 

.16692 

2  .3294 

5-9127 

10 

355i 

56l7 

.16981 

2  .3309 

5-9I57 

20 

35i4 

5505 

.17270 

2  .3323 

5  -9186 

30 

3476 

5393 

•17558 

2  .3337 

5-9215 

40 

3439 

5281 

•17845 

2  -335I 

5  -9245 

50 

34oi 

5168 

.18131 

2  .3365 

5  -9274 

31  oo 

3363 

5°54 

.18416 

2  -3379 

5  -9304 

IO 

3325 

4941 

.18700 

2  .3392 

5  -9334 

20 

3287 

4827 

•18983 

2  .3405 

5  -9363 

30 

3249 

47i3 

.19266 

2  .3418 

5  -9393 

40 

3211 

459s 

.19548 

2  -3431 

5  -9423 

50 

3173 

4483 

.19828 

2  -3444 

5  -9453 

60 

8.509  3134 

8.5114368 

.  20108 

2  .3456 

5  -9484 

TABLES 


357 


TABLE  XIV   (Continued) 


Lat. 

LogA 

LogB 

LogC 

LogD 

LogE 

o       / 

32  oo 

8.5093134 

8.5114368 

.20108 

2  .3456 

5  -9484 

10 

3096 

4252 

.20387 

2.3469 

5  -95I4 

20 

3057 

4i36 

.20666 

2  .3481 

5-9544 

30 

3018 

4020 

.20944 

2  -3493 

5  -9575 

40 

2980 

3903 

.21220 

2  .3504 

5-9605 

50 

2940 

3786 

.21496 

2  .3516 

5  -9636 

33  oo 

2901 

3669 

.21772 

2  -3527 

5  -9667 

10 

2862 

3551 

.22047 

2  -3539 

5.9698 

20 

2823 

3433 

.22321 

2  .3550 

5  -9729 

30 

2784 

3315 

•22594 

2  .3561 

5.9760 

40 

2744 

3*97 

.22866 

2  -357I 

5  -9791 

SO    - 

2704 

3078 

•23138 

2  .3582 

5  -9822 

34  oo 

2665 

2959 

.23409 

2  .3592 

5  -9853 

IO 

2625 

2840 

.23680 

2.3602 

5-9885 

20 

2585 

2720 

•2395° 

2  .3612 

•   5-99i6 

30 

2545 

2600 

.24219 

2  .3622 

5  -9948 

40 

25°5 

2480 

.24488 

2  .3632 

5.9980 

50 

2465 

2360 

.24756 

2  .3642 

6.0011 

35  oo 

2425 

2239 

.25024 

2  -3651 

6.0043 

IO 

2384 

2118 

.25291 

2.3660 

6.0075 

20 

2344 

1997 

•25557 

2.3669 

6  .0107 

30 

2304 

i87S 

•25823 

2  .3678 

6  .0140 

40 

2263 

J754 

.26088 

2.3687 

6.0172 

50 

2222 

1632 

.26353 

2.3695 

6  .0204 

36  oo 

2l82 

1510 

.26617 

2  .3704 

6  .0237 

10 

2141 

1387 

.26881 

2  .37" 

6.0269 

20 

2IOO 

1265 

•27145 

2  .3720 

6  .0302 

30 

2059 

1142 

.27407 

2  .3728 

6  .0334 

40 

20l8 

1019 

.27670 

2  -3735 

6  .0367 

50 

1977 

0895 

•27932 

2  -3743 

6.0400 

37  oo 

IO 

1936 
1895 

0772 
0648 

.28193 
.28454 

2  -375° 
2  .3758 

6.0433 
6  .0466 

20 

1853 

0524 

.28715 

2  .3765 

6  .0499 

30 

1812 

0400 

•28975 

2  .3772 

6  .0533 

40 

1771 

0276 

.29234 

2  -3779 

6.0566 

50 

1729 

0151 

.29494 

2  .3785 

6.0600 

38  oo 

1687 

5.511  0027 

•29753 

2  .3792 

6.0633 

IO 

1646 

8.5109902 

.30011 

2  .3798 

6.0667 

20 

1604 

9777 

.30269 

2.3804 

6  .0701 

30 

1562 

9652 

•30527 

2  .3810 

6.0734 

40 

1521 

9526 

•30785 

2  .3816 

6.0768 

50 

1479 

9401 

.31042 

2  .3822 

6.0802 

60 

8.509  1437 

8.5109275 

.31299 

2.3827 

6.0836 

358 


TABLES 


TABLE  XIV   (Continued) 


Lat. 

Log  A 

LogS 

LogC 

LogD 

Log£ 

o   / 

39  oo 

8.509  U37 

8.5109275 

i  .31299 

2  .3827 

6  .0836 

10 

1395 

9149 

i  ^1555 

2  .3832 

6  .0871 

20 

1353 

9023 

i  .31811 

2  -3838 

6.0905 

3<> 

1311 

8897 

i  .32067 

2  .3843 

6  .0939 

40 

1269 

8771 

i  -32323 

2  .3848 

6  .0974 

50 

1227 

8644 

i  .32578 

2  .3852 

6.1008 

40  oo 

1184 

8517 

i  -32833 

2  .3857 

6.1043 

IO 

1142 

8391 

i  .33088 

2  .3861 

6.1078 

20 

1  100 

8264 

i  .33342 

2  .3866 

6.1113 

30 

1057 

8i37 

i  -33596 

2  .3870 

6.1148 

40 

1015 

8010 

i  -33850 

2  .3874 

6.1183 

50 

0973 

7883 

i  .34104 

2  .3878 

6.1218 

41  oo 

0930 

7755 

i  -34358 

2  .3882 

6.1253 

IO 

0888 

7628 

i  .34611 

2  .3885 

6.1289 

20 

0845 

7500 

i  .34864 

2  -3889 

6.1324 

30 

0803 

7373 

L35II7 

2  .3892 

6.1360 

40 

0760 

7245 

i  -35370 

2  .3895 

6.1395 

50 

0718 

7117 

i  -35623 

2  .3898 

6.1431 

42  oo 

0675 

6989 

i  .35875 

2  .3901 

6.1467 

IO 

0632 

6861 

i  .36127 

2  .3903 

6.1503 

.20 

0590 

6733 

i  -36379 

2  .3906 

6.1539 

30 

0547 

6605 

i  .3663! 

2  .3908 

6.1575 

40 

0504 

6477 

i  .36883 

2  .3910 

6  .1612 

50 

0461 

6348 

i  .37135 

2  .3913 

6.1648 

43  oo 

0419 

6220 

i  .37386 

2  .3914 

6.1684 

10 

0376 

6092 

i  .37638 

2  .3916 

6.1721 

20 

°333 

5963 

i  .37889 

2  .3918 

6.1758 

30 

0290 

5835 

i  .38141 

2  -39*9 

6.1795 

40 

0247 

5706 

i  .38392 

2  -392I 

6.1831 

50 

0204 

5578 

i  -38643 

2  .3922 

6.1868 

44  oo 

0162 

5449 

i  .38894 

2  .3923 

6.1905 

10 

0119 

532o 

i  .39145 

2  .3924 

6.1943 

20 

0076 

5*92 

i  .39396 

2  .3925 

6.1980 

30 

•5090033 

5°63 

i  .39648 

2  ^925 

6  .2017 

40 

.5089990 

4935 

i  .39898 

2  .3926 

6  .2055 

5o 

9947 

4806 

i  .40149 

2  .3926 

6  .2092 

45  oo 

9904 

4677 

1  .40400 

2  .3926 

6  .2130 

10 

9861 

4548 

i  .40651 

2  .3926 

6.2168 

20 

9818 

.4420 

i  .40902 

2  .3926 

6  .2206 

30 

9776 

4291 

I-4II53 

2  .3926 

6  .2244 

40 

9733 

4162 

i  .41404 

2  .3925 

6.2283 

So 

9689 

4034 

i  -41655 

2  .3925 

6.2321 

60 

.508  9647 

•5103905 

i  .41906 

2  .3924 

6.2359 

TABLES 


359 


TABLE  XIV  (Continued) 


Lat. 

Log  A 

Log  B 

LogC 

LogD 

LogE 

o       / 

46  oo 

8.5089647 

8.5I03905 

.41906 

2  .3924 

6  .2359 

10 

9604 

377J 

•42157 

2  .3923 

6  .2398 

20 

956I 

3648 

.42409 

2  .3922 

6  .2436 

30 

95i8 

3519 

.42660 

2  .3921 

6  .2475 

40 

9475 

3391 

.42911 

2  .3920 

6.2514 

50 

9433 

3262 

.43I63 

2  .3918 

6-2553 

47  oo 

939° 

3134 

•43414 

2-39l7 

6.2592 

10 

9347 

3°°5 

.43666 

2  .3915 

6  .2632 

20 

93°4 

2877 

•43917 

2  -3913 

6.2671 

30 

9261 

2749 

.44169 

2.3911 

6.2710 

40 

.    92i9 

2621 

.44421 

2.3909 

6.2750 

50 

9176 

2493 

.44673 

2  .3906 

6  .2790 

48  oo 

9133 

2364 

.44926 

2  .3904 

6.2830 

10 

9091 

2236 

•45!78 

2  .3901 

6.2870 

20 

9048 

2108 

•45431 

2  -3898 

6.2910 

30 

9005 

1981 

•45683 

2  .3895 

6  .2950 

40 

8963 

1853 

•45937 

2.3892 

6.2990 

50 

8920 

I725 

.46190 

2-3889 

6  .3031 

49  oo 

8878 

1598 

.46443 

2  .3886 

6.3071 

10 

8835 

1470 

.46696 

2.3882 

6.3112 

20 

8793 

1343 

•4695° 

2  .3878 

6.3I53 

30 

875° 

1216 

.47204 

2  .3875 

6  .3194 

40 

8708 

1088 

•47459 

2  .3871 

6-3235 

50 

8666 

0962 

•47713 

2  .3866 

6.3276 

50  00 

8623 

0835 

.47968 

2  .3862 

6.3318 

IO 

8581 

0708 

•48223 

2  -3858 

6  -3359 

20 

8539 

0581 

.48478 

2  ^853 

6  .3401 

30 

8497 

0455 

.48734 

2  -3848 

6  -3443 

40 

8455 

0328 

.48989 

2  .3843 

6.3485 

50 

8413 

O202 

.49246 

2  -3838 

6-3527 

51  oo 

837i 

5.5100076 

.49502 

2  .3833 

6.3569 

IO 

8329 

8.5099950 

•49759 

2  .3828 

6.3612 

20 

8287 

9825 

.50016 

2  .3822 

6  .3654 

30 

8245 

9699 

•50273 

2.3817 

6.3697 

40 

8203 

9574 

•50531 

2.3811 

6.3740 

50 

8161 

9448 

.50789 

2  -3805 

6.3782 

52  oo 

8120 

'    9323 

.51048 

2  -3799 

6  .3826 

10 

8078 

9198 

•51307 

2  .3792 

6.3869 

20 

8036 

9074 

.51566 

2.3786 

6.3912 

30 

7995 

8949 

.51826 

2  .3779 

6  .3956 

40 

7953 

8825 

.52086 

2  -3773 

6.4000 

50 

7912 

8701 

•52347 

2  .3766 

6  .4043 

53  oo 

7871 

8577 

.52608 

2  -3759 

6.4088 

IO 

•    7829 

8453 

.52869 

2  -375I 

6.4132 

20 

7788 

8329 

•53i3i 

2-3744 

6.4176 

30 

7747 

8206 

•53393 

2  .3736 

6.4221 

40 

7706 

8083 

•53656 

2  .3729 

6  .4265 

50 

7665 

7960 

•53919 

2.3721 

6.4310 

60 

8.508  7624 

8.509  7838 

1-54183 

2.3713 

6-4355 

TABLES 


-o 

»o  M  t>.  Tf  « 

oo  oo  co  oo  oo 

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NO   NO   NO   NO   NO 

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O    CONO    ON  CNI 

NO   NO   NO   NO   NO 

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to  to  to  to  to 

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CM    CM    CM    CM    CM    CM 

ON  ON  ON  ON  ON  ON 

NO  NO  NO    t^OO 

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CO  co  co  co  co  co 

H    Tf  t>.  ON  CM 

IOOO    M    rj-  t-* 

O    CONO    ON  M 

10  ON  CM    to  ON 

CM    IOOO     CM    tOOO 

H 

00  00  00  00  00 

oo  oo  co  oo  oo 

oo"co"oo'oo"co> 

to  to  to  to  to 

oo  oo  oo  oo  co 

to  to  to  to  to  to 

oo  oo  oo  oo  oo  oo 

3 

tnvo  t^oo  os 

<S   N   N   W   « 

O    M    (N    ro  Tf 

IO\O  Jt^OO  O\ 

«*« 

w*m 

TABLES 


TABLE  XVI.  — COORDINATES  OF  CURVATURE  (METERS) 


Latitudes. 

Long 

26° 

27° 

28° 

29° 

I' 

X 
1668.7 

Y 

O  .1 

X 
1654  -3 

Y 

0  .1 

X 
1639  .4 

Y 

O  .1 

X 
1624  .0 

Y 

0  .1 

2 

3337  -3 

0.4 

3308  .5 

0.4 

3278  .8 

0.4 

3248  .0 

0  -5 

3 

5006.0 

I  .0 

4962  .8 

I  .0 

49l8  .2 

I  .0 

4872  .0 

I  .0 

4 

6674  .6 

I  .-7 

6617.1 

i  .7 

6557  -6 

1.8 

6496. 

1.8 

8343  -3 

2.7 

8271  .4 

2-7 

8197  -o 

2.8 

8120. 

2-9 

6 

IOOII  .9 

3.8 

99257 

3-9 

9836.4 

4.0 

9744. 

7 

11680.6 

5-2 

II579-9 

5-4 

11475.7 

5-5 

11368. 

5^6 

8 

13349  -2 

6.8 

13234-2 

7-° 

7-2 

12992. 

7-3 

9 

15017.9 

8.6 

14888  .5 

8.8 

14754.5 

9-i 

14616.1 

9-3 

10 

16686.6 

10  .6 

16542  .8 

10  .9 

16393  -9 

II  .2 

16240  .1 

ii  -5 

Long. 

30° 

31- 

32° 

33° 

I' 

1608.  i 

O  .1 

I59i  .8 

O.I 

1574.9 

O.I 

1557  .6 

O  .1 

2 

3216.3 

o-5 

3183  -5 

°  -5 

3149  -8 

o-5 

o-5 

3 

4824  .4 

i  .1 

4775  -3 

i  .1 

4724  -8 

i  .1 

4672  .8 

i  .1 

4 

6432  .6 

i-9 

6367  .1 

1.9 

6299.7 

1.9 

6230  .3 

2  .O 

5 

8040.7 

2.9 

795s  -9 

3-o 

7874  -6 

3-0 

7787  .9 

3  -i 

6 

9648  .8 

4.2 

955°  .6 

4-3 

9449-5 

4-4 

9345  -5 

4-4 

7 

11257.0 

5-7 

11142.4 

5-8 

11024.4 

6.0 

10903  .1 

6.0 

8 

12865  .1 

7-5 

12734.2 

7.6 

12599  -4 

.7-8 

12460.7 

7-9 

9 

14473  -2 

9-5 

14325  -9 

9-7 

I4I74.3 

9.8 

14018  .3 

10  .O 

10 

16081.4 

ii  -7 

I59I7.7 

11.9 

15749  -2 

12.  1 

15575  -9 

12.3 

Long. 

34° 

35° 

36° 

37° 

I' 

1539  -8 

O.I 

1521  -5 

O  .1 

1502  .8 

O  .1 

1483  .6 

O  .1 

2 

3079  .6 

°-5 

3043  -o 

o  .5 

3005-5 

0-5 

2967.1 

0.5 

3 

4619  -3 

i  .1 

4564.5 

i  .1 

4508.3 

I  .2 

4450-7 

I  .2 

4 

6159  .1 

2  .O 

6086.0 

2  .O 

6011  .1 

2  .1 

5934-2 

2  .1 

5 

7698.9 

3  •! 

7607.5 

3-2 

7513  .8 

3-2 

7417-8 

3.3 

6 

9238.7 

,  4.5 

9129  .0 

4.6 

9016.6 

4.6 

8901  .4 

47 

I 

10778.5 
12318.3 

6.1 

8.0 

10650  .5 
12172  .0 

6.2 

8.1 

10519  -3 

12022  .1 

5'3 

8.2 

10384  .9 
11868.5 

6-4 
8-3 

9 

13858  .0 

10.  1 

13693  -5 

10.3 

13524  .8 

10.4 

I3352.I 

10.5 

10 

15397  -9 

i2  -5 

15215  -o 

12.7 

15027  .6 

12.8 

14835  -6 

13.0 

362 


TABLES 


TABLE   XVI   (Cow.).  — COORDINATES  OF   CURVATURE  (METERS) 


Latitudes. 

Long. 

38° 

39° 

40° 

41° 

I' 

X 

1463  .9 

y 

O  .1 

X 

1443  -8 

y 

0  .1 

X 
1423  -3 

y 

O  .1 

X 

1402  .3 

y 

O.I 

2 

2927.8 

°-5 

2887  .6 

°-5 

2846  .5 

o-5 

2804  .6 

o-5 

3 

4391  -7 

I  .2 

433i  -4 

I  .2 

4269.8 

I  .2 

4206  .9 

I  .2 

4 

5855  -6 

2  .1 

5775-2 

2  .1 

5693-0 

2  .1 

5609.2 

2  .1 

7319.6 

3-3 

7219.0 

3-3 

7116.3 

3-3 

7011.5 

3-3 

6 

8783  -5 

4.7 

8662  .9 

4-8 

8539  -6 

4.8 

8413  -7 

4.8 

I 

10247  .4 
11711.3 

6.4 
8.4 

10106.7 
ii550-5 

6-5 

8-5 

9962  .8 
11386.1 

6-5 

8-5 

9816  .0 
11218.3 

6.6 
8.6 

9 

I3I75-2 

10  .6 

12994.3 

10.7 

12809  .3 

10.8 

12620  .6 

10.8 

10 

14639.1 

i3-i 

14438.1 

13.2 

14232  .6 

13-3 

14022  .9 

i3-4 

Long. 

42° 

43° 

44° 

45° 

I' 

1380  .9 

O.I 

I359-I 

O.I 

1336.8 

O  .1 

1314.1 

O.I 

2 

2761  .8 

°-5 

2718.1 

°-5 

2673  .6 

0-5 

2628  .3 

0-5 

3 

4142  .7 

I  .2 

4077  .2 

I  .2 

4010  .4 

I  .2 

3942  .5 

I  .2 

4 

5523  -5 

2  .2 

5436.2 

2  .2 

5347  -2 

2  .2 

5256.6 

2  .2 

5 

6904.4 

3-4 

6795  -3 

3-4 

6684  .0 

3-4 

6570.8 

3-4 

6 

8285  .3 

4-8 

8154  -3 

4-9 

8020  .8 

4-9 

7884  .9 

4-9 

7 

9666  .2 

6.6 

95I3-4 

6.6 

9357-7 

6.6 

9199.1 

6.6 

8 

IIO47  -1 

8.6 

10872  .4 

8.6 

10694  .5 

8.6 

10513.2 

8.6 

9 

12428.0 

10.9 

12231.5 

10  .9 

12031  .3 

10  .9 

11827.4 

10  .9 

10 

13808  .8 

13-4 

13590  -5 

!3-5 

13368.1 

J3-5 

13141-5 

I3-S 

Long. 

46° 

47° 

48° 

49°  ' 

I' 

1291  .1 

O  .1 

1267  .6 

O  .1 

1243  -8 

O.I 

1219  .6 

O.I 

2 

2582  .2 

o.S 

2535  -3 

o-5 

2487  .6 

o-S 

2439  -i 

o-5 

3 

3873  -3 

I  .2 

3802  .9 

I  .2 

373i  -4 

I  .2 

3658.7 

I  .2 

4 

5164.4 

2  .2 

5070-5 

2  .2 

4975  -2 

2  .1 

4878  .3 

2  .1 

5 

6455  -5 

3-4 

6338  .2 

3-4 

6219  .0 

3-3 

6097.9 

3-3 

6 

7746  .6 

4.9 

7605.8 

4-8 

7462  .8 

4.8 

73I7-5 

4.8 

7 

9037  .6 

6.6 

8873-5 

6.6 

8706  .6 

6.6 

8537  -o 

6.6 

8 

10328  .7 

8.6 

IOI4I  .1 

8,6 

995°  -4 

8.6 

9756.6 

8.6 

9 

11619  -8 

10.9 

II408.7 

10  .9 

11194.2 

10  .9 

10976  .2 

10.8 

10 

12910  .9 

13-5 

12676.4 

13-5 

12437  .9 

13-4 

I2I95.8 

13-4 

TABLES 


363 


TABLE  XVII.  — COORDINATES  OF  CURVATURE  (METERS) 


Latitudes. 

25° 

30° 

35° 

5° 

X 

5°4  645 

Y 
9  3°7 

X 

482  288 

Y 
10  523 

X 

456  261 

y 

.  ii  421 

IO 

i  008  603 

37  215 

963  658 

42  074 

9"  379 

45  656 

15 
20 

i  511  190 

2  Oil  722 

83  685 
148  656 

i  443  J93 
i  919  982 

94  59i 
167  977 

i  364  214 
i  813  632 

102  619 

182  168 

25 

2  509  518 

232  038 

2  393  116 

262  089 

2  258  507 

284  102 

30 

3  003  900 

333  7i8 

2  861  694 

376  749 

2  697  724 

408  i  68 

Long. 

40° 

45° 

50° 

5° 

426  757 

ii  972 

393  996 

12  160 

358  224 

ii  978 

10 
20 

852  171 
i  274  904 
i  693  628 

47  852 
107  525 
190  805 

786  492 

i  175  994 
i  561  019 

48  594 
109  162 

193  635 

714  847 
i  068  277 
i  416  934 

47  859 
107  482 
190  581 

25 

2  IO7  O23 

297  430 

i  940  103 

301  690 

i  759  262 

296  785 

30 

2  513  790 

427  063 

2  311  802 

432  918 

2  093  731 

425  619 

INDEX 


Aberration,  diurnal,  90,  112 
Absolute  length  of  tape,  39 
Abstract  of  angles,  61 
Accidental  error,  281' 
Accuracy,  of  base  lines,  33 

of  horizontal  angles,  60,  65 

of  latitude,  109 

of  time  observation,  97 
Acetylene  light,  23 
Adding  machine,  243 
Adjustment,  of  level,  241 

of  observations,  283 

of  theodolite,  53 

of  transit,  81 

of  zenith  telescope,  103 
Agar-Baugh,  J.  H.,  32 
Agate,  2ii 
Alidade,  46,  53,  70 
Alignment,  correction,  38 

curve,  143 

Alloy,  32,  63,  213,  240 
Altitude,  73 
Angle,  horizontal,  44 

vertical,  68 
Arcs,  on  earth's  surface,  i 

of  meridian,  134,  187,  181 

oblique,  190 

of  vibration,  220 
Average  error,  316 
Azimuth,  73,  no 

correction,  89 

geodetic,  202 

Back,  azimuth,  166 

reading,  50 
Base  apparatus,  bar,  31 

invar  tape,  28,  32,  34 

steel  tape,  31 
Base  line,  4 

broken,  38 

Epping,  5 

Fire  Island,  5 

Hoi  ton,  31 

marking,  36 

Massachusetts,  5 

measurement  of,  36 

precision  of,  33 

Stanton,  28 


Bench  marks,  243,  345,  249 
Bessel's  spheroid,  158,  193 
Bonne's  projection,  267 
Borda,  222 
Boss's  catalogue,  104 
Bouguer,  233 
Boundaries,  171 
Bowie,  W.,  204 
Box  heliotrope,  22 
Bureau  of  Standards,  40 

California,  triangulation  in,  3,  28,  139 

Catenary,  41 

Center,  of  instrument,  48 

reduction  to,  65 
Centrifugal  force,  208,  235 
Chain  of  triangles,  6,  190 
Check  base,  4 

term,  304,  313 
Chronograph,  78 
Circuits,  electrical,  81,  100 
Circumpolar  star,  83,  no 
Clairaut's  Theorem,  210 
Clarke's  spheroid,  158,  193 
Coefficient,  of  expansion,  32,  39 

of  refraction,  13 
Collimation  error,  56,  57 
Collimator,  vertical,  52 
Compound  events,  280 
Compression  of  earth,  136,  235 
Condition  equations,  294 
Conditions  in  a  figure,  7,  9,  296 
Conformal  projection,  272 
Conic  projection,  272 
Constant  error,  281 
Convergence,  of  meridians,  166 

of  level  surfaces,  245,  253 
Correction,  azimuth;  89 

chronometer,  91 

collimation,  89 

curvature,  112 

level,  87,  113 

rate,  90 

to  observed  quantities,  293 

to  period  of  pendulum,  222 
Correlatives,  304 
Cross  hairs,  46,  48 
Curvature,  correction,  112,  200,  245 


365 


366 


INDEX 


Curvature,  of  earth,  i,  n 

mean,  131 

radius  of,  125,  126,  128 
Curve  of  error,  285 
Curves  on  spheroid,  139 
Cut-off  cylinder,  36 
Cylindrical  projection,  274 

Datum,  249 

Davidson  quadrilaterals,  3 

Declination,  73, 106 

Density  of  earth,  198 

Derrick,  26 

Description  of  station,  61 

Direction,  instrument,  44,  46 
measurement  of,  57 
method  of,  57,  113,  309 
probable  error  of,  6,  65 

Distance,  angles,  7 
check,  8 

Distortion  of  map,  265 

Drag  on  centers,  57 

Dutton,  C.  E.,  202 

Dynamic  number,  255 

Eccentric,  angle,  123 

distance,  65 

station,  65 

Eccentricity  of  circle,  59 
Ecliptic,  73 
Elevation,  253 
Ellipse,  123 
Ephemeris,  74,  103 
Equator,  73 
Equinox,  73 

Equipotential  surface,  185,  252 
Errors,  284 
Exponential  law  of  error,  289 

Figure,  adjustment,  295 

in  triangulation,  6 

of  earth,  185 

strength  of,  6 
Finder  circle,  82 
Flash  apparatus,  214 
Flexure,  of  transit,  74 

of  pendulum  support,  216,  283 
Focus,  82 
Foot  pins,  241 
Forward,  azimuth,  166 

reading,  50 
Function,  precision  of,  322 

Gas  pipe  tower,  26 
Gauss's  method  of  substitutio 
Geodesy  denned,  i 
Geodetic,  datum,  158,  195 


Geodetic,  line,  140 

positions,  158 

surveying,  i 
Geoid,  185,  196,  237 
Gnomonic  projection,  273 
Grade  correction,  37 
Graduation  errors,  57,  58,  59 
Gravitation,  197,  208 

constant,  199 
Gravity,  i,  206 
Great-circle  chart,  273 

track,  274 

Greenwich  catalogue,  104 
Guillaume,  C.  E.,  32 

Hassler,  F.  R.,  268 
Hayford,  J.  F.,  202 
Heliotrope,  n,  19 

box,  22 

Steinheil,  22 
Helmert,  F.  R.,  209 
Horizon,  72 
Hour,  angle,  73 

circle,  73 
Hydrographic  maps,  i,  3 

Illumination,  $7,  78,  215 

Impersonal  micrometer,  76 

Inclination  error,  55 

Interference  bands,  218 

Interferometer,  216 

International  Geodetic  Association,  107, 

237 

Inverse  geodetic  problem,  170 
Isostasy,  202,  235 

Jaderin,  Edw.,  31 
Key  method,  99 

Lambert's  projection,  271 
Laplace,  equation,  201 

stations,  no,  201 
Latitude,  astronomical,  73,  101 

geometric,  123,  135 

geodetic,  123,  160,  175 

reduced,  123,  135 

reduction  to  sea-level,  107 
Law,  of  error,  285 

of  pendulum,  206 
Least  squares,  290 
Legendre's  Theorem,  149,  150 
Level,  correction,  87 

latitude,  101,  103 

pendulum,  213 

precise,  240 

rod,  241 


INDEX 


367 


Level,  stride,  53,  113,  116,  118 

surface,  252 

Longitude,  97,  160,  175 
Loxodrome,  274 

Manometer  tube,  214 
Marking  stations,  61 
Mean  square  error,  316 
Mendenhall,  T.  C.,  211 
Mercator's  projection,  274 
Meridian,  73 

arc,  134,  172 
Meridional  parts,  277 
Micrometer,  46,  48,  101 

transit,  76 

Micrometric  method,  118 
Microscope,  46 

adjustment  of,  54 
Mirror,  19,  20 

back,  23 

size  of,  22 

Missouri  River  Commission,  31 
Mistakes,  283 
Modulus  of  elasticity,  42 

Naval  observatory  time  signals,  220 
New  England  triangulation,  5 
Normal,  123,  186 

equations,  291,  293 

reduced  normal  equations,  303 
North  American  datum,  159 

Observations,  279 
Orthometric,  correction,  254 
elevation,  254 

Parabola,  41 

Parallax,  240 

Parallel  of  latitude,  172,  266 

Pendulum,  206,  211 

Period  of  pendulum,  221 

Personal  equation,  85 

Phase  of  signal,  67 

Pier,  74,  103,  219 

Pivot  inequality,  87 

Plane  coordinates,  174 

curves,  139 
Plane-table  survey,  3 
Plumb  line,  deflection  of,  72,  109,  186, 

Polar  distance,  73 
Pole,  72 

variation,  106 

of  quadrilateral,  298 
Polyconic  projection,  268 
Potential,  250 

energy,  250 


Potential,  function,  250 
Precision  measures,  314 
Primary  triangulation,  2 
Prime  vertical,  73 

component  of  deflection,  201 
Prism,  240 

level,  240 
Probable  error,  317,  325 

of  direction,  6,  65 
Probability,  280 

Quadrilaterals,  6,  8 
Davidson,  3 
adjustment  of,  297 

Ramsden,  44 
Reconnoissance,  n 

for  base,  28 
Reduction,  to  center,  65 

to  station,  257 
Reel  for  tape,  32 
Refraction,  12,  68 

coefficient  of,  13,  259,  261 

horizontal,  59,  139 

differential,  106,  245,  247 
Repeating  instrument,  44 
Repetition,  of  angles,  56,  62 

method,  56,  116 
Residual,  284 
Reticle,  54,  76 
Rhumb  fine,  274 
Right  ascension,  73 
Run  of  micrometer,  49 

Sag  correction,  41 
Scaffold  of  tower,  26 
Sea-level,  determination  of,  249 
reduction  of,  angle  to,  149 

azimuth  to,  120,  136 

base  to,  40 

gravity  to,  233 

latitude,  107,  256 
Secondary  triangulation,  2 
Side,  or  sine,  equation,  299 
Sidereal  time,  73 
Signal  lamp,  24 
Snellius,  i 
Spherical,  coordinates,  278 

excess,  149 
Spheroid,  oblate,  122 
Spheroidal  triangles,  152 
Spring  balance,  34,  43 
Station,  adjustment,  147,  295 
error,  72,  109,  186,  195 
marks,  17,  52 
Stations,  description  of,  16 
Steinheil  heliotrope,  22 


368 


INDEX 


Sterneck,  211 

Strength  of  figure,  4 

Stride  level,  53,  76,  113,  116,  118 

Sub-surface  mark,  36 

Systematic  error,  281 

Talcott's  method,  101,  172 
Temperature,  correction,  39,  222,  248 

errors,  62,  214,  221,  240 

of  rod,  244 

of  tape,  31 
Tension,  apparatus,  34 

correction,  42 
Tertiary  triangulation,  2 
Texas,  triangulation  in,  28,  139 
Thermometers,  for  base  apparatus,  34 
i    for  leveling  rod,  241 

for  pendulum,  213 
Tide  gage,  249 
Topographic,  correction,  234 

deflection,  201 

maps,  i,  3 
Topography,  deflection  of  plumb  line, 

197 
Towers,  n,  25 


Transit,  instrument,  74 

micrometer,  76 
Traverse,  3,  183 
Triangulation,  i,  3 
Tripod,  signal,  18,  26 

of  instrument,  44 
Twist,  of  tripod,  59 

of  triangulation,  62,  202 

Variation  of  pole,  106 
Vernal  equinox,  73 
Vertical,  71 

angles,  61,  68 

circle,  69,  72 

curved,  108,  256 
Vibration  of  towers,  26 

Washington,  triangulation  in,  139 
Weight,  284,  293,  321 
Woodward,  R.  S.,  31 

Zenith,  72 

distance,  73,  101,  IQJS.  acs 
telescope,  101 


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